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Symmetry, Noether and conditional conservation

1 · Prerequisites

2 · Summary

This page develops the conservation laws of the force model of cm02-forces-initial-data-and-newtonian-flow and of the Lagrangian formulation of cm07-stationary-action-and-lagrangian-dynamics as conditional consequences of symmetries of the model's declared presentation. It adds no new physical assumption: Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions defines a symmetry datum $(τ, ξ, B)$ of the mechanical Lagrangian $L = K - U$ by the model's Noether identity, identifies strict data with time-independent $ξ$ with the $C^{2}$ instances of infinitesimal point symmetry of a nonempty chart in the mathematics track, defines the charge $C = (M v) \cdot ξ - (K + U) τ - B$ , and fixes the dimension $μ ℓ^{2} τ^{- 1} σ^{- 1}$ of the charge together with the convention that $ξ$ generates the configuration change $x \mapsto x + s ξ$ . The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate proves the point theorem of the model: the charge of a strict symmetry with time-independent generator is constant along every motion of the global potential branch, and for an arbitrary presentation its derivative equals the residual force paired with the generator, so conservation holds exactly when that pairing vanishes. The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p \cdot ξ - E τ - B$ , its residual-force rate and its potential gauge is the general theorem of the page: a datum with time reparametrisation has the charge $(M v) \cdot ξ - E τ - B$ conserved in the global potential branch, with the $τ = 0$ and $ξ = 0, τ = 1$ cases recovering the strict point charge and minus the mechanical energy, with the residual-force rate $⟨ d, ξ - v τ ⟩$ , and with the potential gauge: replacing $U$ by $U + c (t)$ and $B$ by $B - c τ$ preserves the identity and leaves the charge unchanged.

The concrete charges are then proved against explicit hypotheses on the declared potential. Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential shows that the time-translation datum, with its parameter measured in seconds, satisfies the model identity exactly for autonomous representatives, proves conservation of $K + U$ in the autonomous global potential branch, and, in that branch, exhibits the failure of conservation along an actual maximal motion whenever $\partial_{t} U$ is nonzero at a state. Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness uses a dimensionless direction vector and a length-valued parameter to convert the differential condition $\sum_{i} a \cdot \nabla_{x_{i}} U = 0$ , equivalently invariance of $U$ along simultaneous translation segments contained in $Q$ , into conservation of the total-momentum component $a \cdot P$ in the global potential branch, with the failure rate of the sharpness clause. Comparing endpoints across excluded segments additionally requires agreement of the component constants specified in that theorem. Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin does the same for the rotation family with dimensionless axis vector $a$ and radian-valued parameter in that branch, whose charge is the component $a \cdot J$ of the total angular momentum about the inertial origin, together with its full vector form for rotationally invariant potentials; and The Galilean boost charge of the model: exact condition, uniform centre-of-mass motion, the induced motions and the boundary of the conservation law treats the Galilean boost datum with dimensionless direction $a$ and velocity-valued family parameter, whose charge has mass-length units and is conserved under the differential translation condition in the global potential branch. It records that an admissible boosted curve is a motion exactly when the potential gradients at its paired endpoints agree; this follows when the joining segments lie in $Q$ or the full endpoint-invariance criterion holds, gives the general rate $K_{a}^{'} = t a \cdot F_{t o t}$ , and marks the boundary: Galilean frame availability is kinematic, while the boost charge is a property of the model's Lagrangian. The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions supplies the bookkeeping of a time-dependent coordinate description: the chart energy function differs from the inertial mechanical energy by $⟨ X_{t}, M x^{'} ⟩$ , is conserved along chart solutions when the substituted Lagrangian has no explicit time dependence, and the chart description of every motion of the global potential branch solves that chart equation. The page closes with Scope of the symmetry-conservation statements: force laws that break kinematic symmetries, residual forces, moving coordinates and the empirical boundary: in the global potential branch, uniform fields, confining or time-dependent potentials can break the corresponding symmetries, residual forces destroy charges at the computed rates, "the Hamiltonian is conserved" is not a statement about the inertial mechanical energy in a moving chart, no converse from conservation to symmetry is asserted, and every conservation statement is a conditional deduction of the model rather than an empirical finding.

Worked checks live on cm08-symmetry-noether-and-conditional-conservation-examples: the uniform-force model with its partial translation charges and its exact failure rates, a pair-potential model whose translation and rotation charges are verified symbolically and on an explicit rigid-rotation motion, an oscillator observed in a uniformly rotating chart whose conserved chart Hamiltonian differs from the inertial mechanical energy, and a driven model with a time-dependent potential whose energy changes at the rate of the explicit time dependence.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions

Physical scope

The symmetry vocabulary of the force model with a declared potential presentation in one inertial frame: symmetry data as triples (tau, xi, B) satisfying the Noether identity for the mechanical Lagrangian K - U of that presentation, which carries only the potential part when residual forces are present, strict data and their exact relation to the chart-level infinitesimal point symmetries, the charge M v . xi - (K+U) tau - B together with its dimension and its behaviour under rescaling of the family parameter, and the status of the notion. The item defines only; it asserts no conservation law, no existence of any symmetry of a given model, and no empirical content.

Definition

Adopt one force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N \geq 1$ particle labels with fixed masses $m_{1}, \dots, m_{N} > 0$ (clause (i)), an open nondegenerate time window $I_{0} \subseteq R$ , an open configuration domain $Q \subseteq R^{3 N}$ and a force map $F$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate, and fix one inertial frame with adapted coordinates $(t, x)$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). Adopt in addition a potential presentation $(U, d)$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a function $U : I_{0} \times Q \to R$ of class $C^{2}$ of dimension $μ ℓ^{2} τ^{- 2}$ with $F_{i} (t, x, v) = - \nabla_{x_{i}} U (t, x) + d_{i} (t, x, v)$ , with residual forces $d = (d_{1}, \dots, d_{N})$ of the standing regularity. Let $L = K - U$ , $O : = I_{0} \times Q \times R^{3 N}$ and $M$ be the mechanical Lagrangian of the presentation, the chart domain and the mass operator of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, so that $L$ is of class $C^{2}$ on the open set $O$ . When $d \equiv̸ 0$ , $L$ represents only the potential part and excludes the residual forces; its symmetry identity places no condition on $d$ and does not assert a symmetry of the full force map. When $d \equiv 0$ , $L$ is a Lagrangian of the model itself. When $Q \neq \emptyset$ , $(O, L)$ is a time-dependent Lagrangian chart over every $[a, b] \times Q$ with $a < b$ and $[a, b] \subseteq I_{0}$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, inner products and norms are Euclidean (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and differentiability classes are those of $C^{k}$ Euclidean maps and diffeomorphisms. For a $C^{1}$ function $φ$ on an open subset of $R \times R^{3 N}$ and $(t, x, v) \in O$ write

$D_{t} φ (t, x, v) : = φ_{t} (t, x) + φ_{x} (t, x) \cdot v,$

the derivative of $s \mapsto φ (s, x + v (s - t))$ at $s = t$ ; the last two slots of $L$ are $(x, v)$ , and $L_{t}$ , $L_{x}$ , $L_{v}$ are the gradients of $L$ in its three slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

(a) Symmetry datum. A symmetry datum of the mechanical Lagrangian of the declared potential presentation is a triple $(τ, ξ, B)$ of maps

$τ : I_{0} \times Q \to R, ξ : I_{0} \times Q \to R^{3 N}, B : I_{0} \times Q \to R,$

of class $C^{2}$ , that satisfies for every $(t, x, v) \in O$ the Noether identity of the model:

$L_{t} (t, x, v) τ (t, x) + L_{x} (t, x, v) \cdot ξ (t, x) + L_{v} (t, x, v) \cdot (D_{t} ξ (t, x, v) - v D_{t} τ (t, x, v)) + L (t, x, v) D_{t} τ (t, x, v) = D_{t} B (t, x, v) .$

Because $K (v) = \frac{1}{2} v \cdot M v$ does not depend on $t$ or $x$ and $U$ does not depend on $v$ , the identity is the following identity among the model's data:

$- U_{t} (t, x) τ (t, x) - \nabla_{x} U (t, x) \cdot ξ (t, x) + (M v) \cdot (D_{t} ξ (t, x, v) - v D_{t} τ (t, x, v)) + (K (v) - U (t, x)) D_{t} τ (t, x, v) = D_{t} B (t, x, v) .$

It involves the declared potential $U$ , the masses through $M$ , and the boundary function $B$ ; it is an identity imposed on the model's declared data on the whole state domain $O$ .

(b) Strict data and point symmetries. A symmetry datum is strict when $τ \equiv 0$ ; its identity reads $L_{x} \cdot ξ + L_{v} \cdot D_{t} ξ = D_{t} B$ . The following chart identification is for $Q \neq \emptyset$ . If a strict datum has $ξ$ independent of $t$ , then the pair $(ξ, B)$ is an infinitesimal point symmetry of the chart $(O, L)$ in the sense of Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge: that definition's identity, instantiated with $n = 3 N$ in the configuration slot, is verbatim $L_{x} (t, x, v) \cdot ξ (x) + L_{v} (t, x, v) \cdot D ξ (x) v = B_{t} (t, x) + B_{x} (t, x) \cdot v$ . Conversely every infinitesimal point symmetry of the chart $(O, L)$ whose generator and boundary function are of class $C^{2}$ , with the boundary function defined on $I_{0} \times Q$ , is a strict datum with $ξ$ independent of $t$ . A general datum is a point symmetry with time reparametrisation: the family

$x_{i} ⟼ x_{i} + s ξ_{i} (t, x), t ⟼ t + s τ (t, x)$

changes the coordinate velocity to first order by $D_{t} ξ - v D_{t} τ$ , and the identity says that the first-order change of the action density $L d t$ , expressed per unit of the original $d t$ , is $D_{t} B$ . The term $L D_{t} τ$ accounts for the change of the time differential; the variation of $L$ alone equals $D_{t} B$ only when this term vanishes.

(c) Charge. The charge of the datum is the function

$C : O \to R, C (t, x, v) : = L_{v} (t, x, v) \cdot ξ (t, x) - E (t, x, v) τ (t, x) - B (t, x),$

where

$E (t, x, v) : = L_{v} (t, x, v) \cdot v - L (t, x, v) = K (v) + U (t, x)$

is the mechanical energy function of the chart (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Since $L_{v} = M v$ ,

$C (t, x, v) = (M v) \cdot ξ (t, x) - (K (v) + U (t, x)) τ (t, x) - B (t, x) .$

(d) Dimensions and rescaling. Read the family parameter $s$ as carrying a declared unit $σ$ (for instance radians for a rotation family, seconds for a time-reparametrisation family, or a dimensionless parameter). Then $τ$ has dimension $τ σ^{- 1}$ , every component $ξ_{i}$ has dimension $ℓ σ^{- 1}$ , $B$ has dimension $μ ℓ^{2} τ^{- 1} σ^{- 1}$ (action per parameter unit) and $C$ has the same dimension as $B$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Replacing $(τ, ξ, B)$ by $(c τ, c ξ, c B)$ with a real constant $c \neq 0$ — the same family reparametrised by $s \mapsto c s$ — rescales $C$ by $c$ .

(e) Status. This item fixes the model's symmetry vocabulary: which data satisfy the model's Noether identity, and which function is their charge. It asserts neither that a given model admits a nonzero symmetry datum, nor that the charge of a datum is conserved. Conservation is proved under explicit hypotheses by the theorems of this page, and the hypotheses must be verified against the declared potential $U$ and the masses: a change of inertial frame, the affine structure of the spacetime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, or a relabelling of the particle labels is a kinematic or descriptive operation and is not by itself a symmetry datum of $L$ . Every statement that uses a charge names the potential representative used (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)); the effect of changing the representative is recorded in the general Noether theorem of this page.

Remarks

Generator convention. The sign of $ξ$ is fixed by the family $x \mapsto x + s ξ$ : with an inverse-family convention, in which the transformed coordinate is written as a function of the old one by $x \mapsto x - s ξ$ , the same identity holds with $C$ replaced by $- C$ . The present convention gives $a \cdot J$ for the rotation generator $ξ_{i} = a \times x_{i}$ . In SICM Section 1.8.5, the rotation matrix in Eq. (1.163) has derivative $(- y, x, 0)$ at zero, whereas Eq. (1.168) displays $(y, - x, 0)$ and hence the negative angular momentum. Those displayed formulas have a sign inconsistency; the coordinate formula and the fixed generator, rather than that printed derivative, determine the sign used here.
Relation to the mathematical track. The chart-level results Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge, The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions and Noether identity for point transformations with time reparametrisation: the conserved charge $p \cdot ξ - E τ - B$ concern a general chart Lagrangian; this item only instantiates them at the mechanical Lagrangian of the declared presentation, and the identity above is exactly theirs written in model terms. The conservation theorems of this page are instances of those results, not independent postulates.
What a symmetry is not. The identity constrains the potential: for the translation family with frozen generator $ξ = a$ and $B = 0$ it fails at the state $(t, x, v)$ exactly where $a \cdot \nabla_{x} U (t, x) \neq 0$ , and for the quadratic potential $U = \frac{k}{2} ∣ x ∣^{2}$ with $k > 0$ it fails exactly where $a \cdot x \neq 0$ , since $a \cdot \nabla_{x} U = k a \cdot x$ . A model may therefore possess the full kinematic group of Galilean frame changes and still admit no translation symmetry datum. The worked models of the companion page make this failure explicit.
Not asserted. No converse is asserted: from the conservation of a quantity along the motions of a model it does not follow that the model has a symmetry datum producing it, and no claim is made that the conservation laws observed in nature are generated by symmetries. The item also does not treat discrete symmetries, gauge families whose generators are not $C^{2}$ , field-theoretic actions, or global existence of the transformation families; only the identity and the charge are defined here.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate

Physical scope

The point Noether theorem of the force model with a declared potential presentation in one inertial frame: for a strict symmetry datum of the mechanical Lagrangian that is independent of time, the charge M x' . xi - B is constant along every motion of the global potential branch; for an arbitrary presentation the charge of a strict datum has derivative equal to the residual force paired with the generator, so conservation holds exactly when that pairing vanishes along the motion. The theorem is conditional on the adopted force branch, the declared presentation, the symmetry identity and, in the conservation clause, on the absence of residual forces; it asserts no symmetry of any particular force law and carries no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, and let $(τ, ξ, B)$ be a symmetry datum of the model's mechanical Lagrangian with $τ \equiv 0$ (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions clause (a)); write

$C (t, x, v) : = (M v) \cdot ξ (t, x) - B (t, x) ((t, x, v) \in O)$

for its charge. Then:

(conservation in the global potential model). If $d \equiv 0$ and $ξ$ is independent of $t$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto C (t, x (t), x^{'} (t))$ is constant on $J$ .
(residual-force rate). For an arbitrary presentation $(U, d)$ , along every motion $x : J \to Q$ the function $t \mapsto C (t, x (t), x^{'} (t))$ is of class $C^{1}$ on $J$ and

$C^{'} (t) = ⟨ d (t, x (t), x^{'} (t)), ξ (t, x (t)) ⟩ (t \in J),$

where the pairing is the Euclidean inner product of $R^{3 N}$ ; consequently $C$ is constant on $J$ if and only if this pairing vanishes for every $t \in J$ .

Clause 1 is the conditional conservation law of the model's global potential branch under a symmetry identity of its Lagrangian; clause 2 quantifies the exact rate at which a residual force destroys it. Neither clause asserts that any model possesses a strict symmetry datum, and neither carries empirical content.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the strict symmetry datum $(0, ξ, B)$ and the motion $x : J \to Q$ of the statement.

[F1]

Model data and chart (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $O = I_{0} \times Q \times R^{3 N}$ is open and $(O, L)$ is a time-dependent Lagrangian chart whenever $Q \neq \emptyset$ and $a < b$ with $[a, b] \subseteq I_{0}$ , over $[a, b] \times Q$ ; $L (t, x, v) = K (v) - U (t, x)$ with $K (v) = \frac{1}{2} v \cdot M v$ , $L_{v} (t, x, v) = M v$ , $L_{x} (t, x, v) = - \nabla_{x} U (t, x)$ and $L_{t} (t, x, v) = - U_{t} (t, x)$ ; the force map is $F_{i} = - \nabla_{x_{i}} U + d_{i}$ .

[F2]

The symmetry datum and its charge (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions, Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge): $ξ : I_{0} \times Q \to R^{3 N}$ and $B : I_{0} \times Q \to R$ are of class $C^{2}$ and satisfy the strict identity $L_{x} (t, x, v) \cdot ξ (t, x) + L_{v} (t, x, v) \cdot D_{t} ξ (t, x, v) = D_{t} B (t, x, v)$ on $O$ ; the charge is the displayed function $C$ ; when $ξ$ is independent of $t$ the pair $(ξ, B)$ is an infinitesimal point symmetry of the chart $(O, L)$ in the sense of the cited definition.

[F3]

The mathematical conservation theorem (The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions, Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression): for an infinitesimal point symmetry $(ξ, B)$ of a time-dependent Lagrangian chart and a $C^{2}$ curve $q : I \to U$ on an open interval $I$ with graph in the chart domain that solves the Euler-Lagrange equation $E_{L} (q) = 0$ , the function $t \mapsto L_{v} \cdot ξ - B$ is constant on $I$ .

[F4]

The Euler-Lagrange equivalence in model terms (The stationary-action criterion and the potential-part Euler-Lagrange equation, Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch)): for every $C^{2}$ admissible curve $z$ of the chart instance and reals $a < b$ with $[a, b] \subseteq I_{0}$ , stationarity for the action, $E_{L} (z) = 0$ on $[a, b]$ and $M z^{''} (t) = - \nabla_{x} U (t, z (t))$ on $[a, b]$ are equivalent; and every motion of the model with $d \equiv 0$ restricts on such an interval to a curve satisfying these conditions.

[F5]

Motions (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance): a motion $x : J \to Q$ on an interval with nonempty interior is a $C^{2}$ curve with $J \subseteq I_{0}$ and $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), x^{'} (t))$ for all $t \in J$ , i.e. $M x^{''} = - \nabla_{x} U + d$ along the motion.

[F6]

Calculus along the motion (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant): the curves $t \mapsto ξ (t, x (t))$ and $t \mapsto B (t, x (t))$ are $C^{1}$ with derivatives $D_{t} ξ (t, x (t), x^{'} (t))$ and $D_{t} B (t, x (t), x^{'} (t))$ ; sums and pairings of differentiable curves differentiate by the product rule; a $C^{1}$ real function on an interval is constant if and only if its derivative vanishes at every point.

[F7]

Regularity of the data ( $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): $ξ$ and $B$ are $C^{2}$ , $U$ is $C^{2}$ and a motion is $C^{2}$ , so all composites below are at least $C^{1}$ .

Proof

technique · direct; convert a motion into an Euler-Lagrange solution for the conservation clause and instantiate the mathematical Noether theorem, then differentiate the charge along the motion for the residual-force rate

1.1F1F4given

Clause 1, motion as an Euler-Lagrange solution: let $x : J \to Q$ be a motion with $d \equiv 0$ . The interior $J^{\circ}$ is a nonempty open interval. For any $t_{0} \in J^{\circ}$ choose $a < b$ with $t_{0} \in (a, b)$ and $[a, b] \subseteq J^{\circ}$ ; the restriction $x ∣_{[a, b]}$ is a $C^{2}$ admissible curve of the chart instance and is stationary for the action of $L$ by [F4], so it satisfies $E_{L} (x) = 0$ on $[a, b]$ . As this holds for each $t_{0} \in J^{\circ}$ , $E_{L} (x) = 0$ throughout $J^{\circ}$ .

1.2F2F5F6F7algebra

Clause 2, differentiation: along any motion, $t \mapsto C (t, x (t), x^{'} (t))$ is of class $C^{1}$ by [F6] and [F7], and differentiating $C = (M x^{'}) \cdot ξ (t, x) - B (t, x)$ gives $C^{'} = (M x^{''}) \cdot ξ + (M x^{'}) \cdot D_{t} ξ - D_{t} B$ at each $t \in J$ . Substituting the strict identity of [F2] with $(t, x, v) = (t, x (t), x^{'} (t))$ , i.e. $D_{t} B = - \nabla_{x} U \cdot ξ + (M x^{'}) \cdot D_{t} ξ$ , and then $M x^{''} = - \nabla_{x} U + d$ from [F5], gives $C^{'} = (M x^{''}) \cdot ξ + \nabla_{x} U \cdot ξ = (M x^{''} + \nabla_{x} U) \cdot ξ = ⟨ d (t, x (t), x^{'} (t)), ξ (t, x (t)) ⟩ .$

2.1F2F3F7step 1.1

Clause 1, conservation: by [F2] the pair $(ξ, B)$ , with $ξ$ independent of $t$ , is an infinitesimal point symmetry of the chart $(O, L)$ . Step 1.1 shows that $x ∣_{J^{\circ}}$ is a $C^{2}$ curve on the open interval $J^{\circ}$ with graph in $O$ solving $E_{L} = 0$ , so the mathematical conservation theorem [F3] applies there and makes $t \mapsto L_{v} (t, x, x^{'}) \cdot ξ (x) - B (t, x) = (M x^{'} (t)) \cdot ξ (x (t)) - B (t, x (t))$ constant on $J^{\circ}$ . The charge is continuous on $J$ by [F7]; every endpoint of $J$ that belongs to $J$ is a limit of points of $J^{\circ}$ , so continuity gives the same constant value there. Thus the charge is constant on $J$ .

2.2F6step 1.2

Clause 2, conservation criterion: by step 1.2 the derivative of $t \mapsto C (t, x (t), x^{'} (t))$ is the pairing displayed there; by [F6] a $C^{1}$ function on the interval $J$ is constant on $J$ if and only if its derivative vanishes everywhere on $J$ , so $C$ is constant on $J$ exactly when the pairing vanishes for every $t \in J$ .

3.1step 2.1step 2.2∎

Clause 1 is steps 1.1 and 2.1, and clause 2 is steps 1.2 and 2.2; both clauses are conditional deductions from the indicated hypotheses, and no claim of existence of a symmetry datum or of empirical content is made.

Remarks

Where the identity is used. Clause 1 uses the strict identity on the whole chart through the mathematical point-symmetry theorem; clause 2 uses the same identity only at the graph points of the motion and evaluates the motion equation. For a presentation with $d \equiv 0$ the two routes agree, and clause 2 then gives $C^{'} = 0$ directly; the field-theoretic reading is that the potential part of the force does no work against the generator, while the residual part does, at rate $⟨ d, ξ ⟩$ . The specialisation of this statement with $ξ = ξ (x)$ to the mathematical case The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions is the reason the framework reuses the chart-level notion of symmetry as its model notion.
Conditional conservation. The pairing criterion makes the conditions explicit: with residual forces the charge of a strict symmetry is conserved exactly if the residual force is orthogonal to the generator along the motion. If the total pairing $⟨ d, ξ ⟩$ is nonzero at a point of the motion — for instance for a nonzero friction force paired with a translation generator — the corresponding charge is not constant on that motion.
What is not claimed. No converse is asserted: from the constancy of $(M x^{'}) \cdot ξ - B$ along the motions one cannot conclude that the identity holds, and the existence of a symmetry datum for a given model is a separate verification against its declared $U$ and masses. The statement is also not an empirical law: it concerns the model's motions under the adopted force branch and the declared presentation.

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The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p \cdot ξ - E τ - B$ , its residual-force rate and its potential gauge

Physical scope

The general point Noether theorem of the force model with a declared potential presentation in one inertial frame: for a symmetry datum with time reparametrisation, the charge M x' . xi - (K+U) tau - B is constant along every motion of the global potential branch; the charge has derivative equal to the residual force paired with xi - x' tau for an arbitrary presentation; the tau=0 and xi=0, tau=1 specialisations reduce to the strict point-symmetry charge and to minus the mechanical energy; and replacing the potential representative U by U+c(t) with the boundary function B-c tau preserves the identity and leaves the charge unchanged. The theorem is conditional on the adopted force branch, the declared presentation and the symmetry identity; it asserts no symmetry of any particular force law, no converse from conservation to symmetry, and carries no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, and let $(τ, ξ, B)$ be a symmetry datum of the model's mechanical Lagrangian (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions clause (a)), with charge

$C (t, x, v) : = (M v) \cdot ξ (t, x) - (K (v) + U (t, x)) τ (t, x) - B (t, x) ((t, x, v) \in O) .$

Then:

(conservation in the global potential model). If $d \equiv 0$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto C (t, x (t), x^{'} (t))$ is constant on $J$ .
(specialisations). At $τ \equiv 0$ the model identity is the strict identity of clause (b) of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions and the charge reduces to $(M v) \cdot ξ - B$ ; at $ξ \equiv 0$ , $τ \equiv 1$ , $B \equiv 0$ the model identity holds on $O$ if and only if $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ , and the charge is $- E$ , where $E = K + U$ is the model's mechanical energy function.
(residual-force rate). For an arbitrary presentation $(U, d)$ , along every motion $x : J \to Q$ the function $t \mapsto C (t, x (t), x^{'} (t))$ is of class $C^{1}$ on $J$ and

$C^{'} (t) = ⟨ d (t, x (t), x^{'} (t)), ξ (t, x (t)) - x^{'} (t) τ (t, x (t)) ⟩ (t \in J);$

consequently $C$ is constant on $J$ if and only if this pairing vanishes for every $t \in J$ .

(potential gauge). Let $c : I_{0} \to R$ be of class $C^{2}$ and let $U^{'} : = U + c$ be the representative of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c), with Lagrangian $L^{'} = K - U^{'}$ ; put $B^{'} : = B - c τ$ . Then $(τ, ξ, B^{'})$ is a symmetry datum of the model's mechanical Lagrangian for the representative $U^{'}$ if and only if $(τ, ξ, B)$ is one for $U$ , and its charge is the same function $C$ . In particular the conserved quantity of clause 1 does not depend on the choice of potential representative, although the mechanical energy $E = K + U$ does.

All clauses are conditional deductions from the adopted force branch, the declared presentation and the stated symmetry identity; no clause asserts that a given model admits a symmetry datum, and no clause carries empirical content.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the symmetry datum $(τ, ξ, B)$ , the motion $x : J \to Q$ and, in clause 4, the function $c$ .

[F1]

Model data and chart (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $O = I_{0} \times Q \times R^{3 N}$ is open and $(O, L)$ is a time-dependent Lagrangian chart whenever $Q \neq \emptyset$ and $a < b$ with $[a, b] \subseteq I_{0}$ , over $[a, b] \times Q$ ; $L = K - U$ with $K (v) = \frac{1}{2} v \cdot M v$ , $L_{v} = M v$ , $L_{x} = - \nabla_{x} U$ , $L_{t} = - U_{t}$ ; the force map is $F_{i} = - \nabla_{x_{i}} U + d_{i}$ ; and $U^{'} = U + c$ presents the same force map with the same residual forces, with $L^{'} = L - c$ .

[F2]

Symmetry datum and charge (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): the maps $τ, ξ, B$ are of class $C^{2}$ and satisfy the model Noether identity $L_{t} τ + L_{x} \cdot ξ + L_{v} \cdot (D_{t} ξ - v D_{t} τ) + L D_{t} τ = D_{t} B$ on $O$ , and the charge is the displayed function; at $τ \equiv 0$ the identity is the strict one and the charge is $(M v) \cdot ξ - B$ .

[F3]

The mathematical general Noether theorem (Noether identity for point transformations with time reparametrisation: the conserved charge $p \cdot ξ - E τ - B$ ): for a time-dependent Lagrangian chart, $C^{2}$ maps $τ, ξ, B$ satisfying the same identity on the chart domain, and a $C^{2}$ curve that solves the Euler-Lagrange equation, the function $t \mapsto L_{v} \cdot ξ - E_{L} τ - B$ is constant along the curve, where $E_{L} = L_{v} \cdot v - L$ ; its clause 1 identifies the case $τ = 0$ with the infinitesimal point symmetry identity and charge when $ξ$ is independent of time; for time-dependent $ξ$ , $τ = 0$ still gives the model's strict identity, and the general conservation conclusion still applies, and its clause 2 states that at $ξ = 0$ , $τ = 1$ , $B = 0$ the identity holds if and only if $L_{t} = 0$ on the chart domain, with charge $- L_{v} \cdot v + L$ .

[F4]

Motions and the Euler-Lagrange equivalence (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, The stationary-action criterion and the potential-part Euler-Lagrange equation, Stationary curves of the mechanical action are exactly the restrictions of the model's motions (global potential branch)): a motion $x : J \to Q$ is a $C^{2}$ curve on an interval with nonempty interior, endpoints understood with one-sided derivatives, satisfying $M x^{''} = - \nabla_{x} U + d$ on $J$ ; only when $d \equiv 0$ does every restriction to a closed subinterval $[a, b] \subseteq J$ with $a < b$ satisfy $E_{L} (x) = 0$ and stationarity for the action of $L$ . For arbitrary residual forces, no stationarity or Euler-Lagrange equation for $L = K - U$ is asserted.

[F5]

Energy balance (Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form clause 1): along a motion $x : J \to Q$ of the presentation $(U, d)$ , the mechanical energy satisfies $E^{'} (t) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t, x (t), x^{'} (t)), x_{i}^{'} (t) ⟩$ for every $t \in J$ , the evaluated residuals being those of the statement.

[F6]

Calculus along the motion (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant): composites $t \mapsto φ (t, x (t))$ of $C^{1}$ maps with the motion are $C^{1}$ with derivative $D_{t} φ$ ; sums, products and pairings differentiate by the usual rules; a $C^{1}$ real function on an interval is constant if and only if its derivative vanishes everywhere.

[F7]

Regularity and dimensions ( $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): all maps occurring in the proof are of the stated differentiability class, and $⟨ \cdot, \cdot ⟩$ is the Euclidean inner product of $R^{3 N}$ .

Proof

technique · direct; obtain clause 1 from the mathematical general Noether theorem applied to the Euler-Lagrange description of a motion, clauses 2 and 4 by evaluating the model identity under the stated substitutions, and clause 3 by differentiating the charge along the motion

1.1F1F4given

Clause 1, motion as an Euler-Lagrange solution: let $x : J \to Q$ be a motion with $d \equiv 0$ . Its interior $J^{\circ}$ is a nonempty open interval. For any $t_{0} \in J^{\circ}$ choose $a < b$ with $t_{0} \in (a, b)$ and $[a, b] \subseteq J^{\circ}$ ; by [F4] the restriction $x ∣_{[a, b]}$ is stationary for the action of $L$ and satisfies $E_{L} (x) = 0$ on $[a, b]$ . Hence $E_{L} (x) = 0$ throughout $J^{\circ}$ .

1.2F1F2F4F5F6F7algebra

Clause 3, differentiation: along a motion $x : J \to Q$ , the curve $t \mapsto C (t, x (t), x^{'} (t))$ is $C^{1}$ by [F6] and [F7], and differentiating $C = (M x^{'}) \cdot ξ - E τ - B$ gives $C^{'} = (M x^{''}) \cdot ξ + (M x^{'}) \cdot D_{t} ξ - E^{'} τ - E D_{t} τ - D_{t} B .$ Substituting $E^{'} = \partial_{t} U + ⟨ d, x^{'} ⟩$ from [F5], the model identity of [F2] evaluated at the graph point, $D_{t} B = - U_{t} τ - \nabla_{x} U \cdot ξ + (M x^{'}) \cdot (D_{t} ξ - x^{'} D_{t} τ) + (K - U) D_{t} τ,$ and $M x^{''} = - \nabla_{x} U + d$ from [F4], the terms $(M x^{'}) \cdot D_{t} ξ$ , $U_{t} τ$ and $E D_{t} τ$ cancel against the corresponding terms of the identity, and the identity $⟨ M x^{'}, x^{'} ⟩ = 2 K$ together with $E = K + U$ leaves $C^{'} = (M x^{''} + \nabla_{x} U) \cdot ξ - ⟨ d, x^{'} ⟩ τ = ⟨ d, ξ - x^{'} τ ⟩ .$

1.3F1F2F7algebra

Clause 4, preservation of the identity: write $φ : = c τ$ and compare the two sides of the model identity for the replaced data $(U^{'}, B^{'})$ with those for $(U, B)$ at a point of $O$ : since $L^{'} = L - c$ by [F1], the left-hand side changes by $- c_{t} τ - c D_{t} τ = - D_{t} φ$ , because the identity is linear in $L$ and its first derivatives and $c$ is independent of $x$ and $v$ ; the right-hand side is $D_{t} B - D_{t} φ$ ; hence the replaced identity holds on $O$ if and only if the original one does.

1.4F1F2F3algebra

Clause 2, specialisations: setting $τ \equiv 0$ in [F2] gives $D_{t} τ = 0$ , so directly the model identity becomes $L_{x} \cdot ξ + L_{v} \cdot D_{t} ξ = D_{t} B$ , the strict identity of clause (b) of the symmetry definition, and its charge reduces to $(M v) \cdot ξ - B$ ; by clause 2 of [F3] and $L_{t} = - U_{t}$ from [F1], the datum $ξ \equiv 0$ , $τ \equiv 1$ , $B \equiv 0$ satisfies the identity on $O$ if and only if $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ , and its charge is $- ((M v) \cdot v - L) = - (K + U) = - E$ .

2.1F1F2F3F7step 1.1

Clause 1, conservation: by [F2] the datum $(τ, ξ, B)$ satisfies the model identity on $O$ , which is the identity of [F3]. Step 1.1 shows that the motion restricted to the open interval $J^{\circ}$ has graph in $O$ and solves $E_{L} = 0$ , so [F3] applies with $I : = J^{\circ}$ and gives that $t \mapsto (M x^{'} (t)) \cdot ξ (t, x (t)) - E_{L} (t, x, x^{'}) τ (t, x (t)) - B (t, x (t))$ is constant on $J^{\circ}$ , where $E_{L} = L_{v} \cdot v - L = K + U$ by [F1]; this is exactly $t \mapsto C (t, x (t), x^{'} (t))$ . The charge is continuous on $J$ by [F7]; each included endpoint is a limit of points of $J^{\circ}$ , so the same value holds there. Thus the charge is constant on $J$ .

2.2F1F2step 1.3algebra

Clause 4, the charge: for the replaced data the charge is $(M v) \cdot ξ - (K + U + c) τ - (B - c τ) = (M v) \cdot ξ - (K + U) τ - B = C$ , because the two terms containing $c$ cancel; combined with step 1.3, this proves clause 4.

3.1step 1.2step 1.4step 2.1step 2.2∎

Clause 2 is step 1.4; clause 3 is step 1.2 together with the elementary equivalence, for a $C^{1}$ function on the interval $J$ , between constancy and vanishing derivative; clauses 1 and 4 are steps 2.1 and 2.2; all clauses are conditional deductions from the stated hypotheses, and no converse from conservation to symmetry, no existence of a symmetry datum and no empirical statement is asserted.

Remarks

The three charges of the framework. Clause 2 records that the general identity contains the strict point-symmetry charge $(M v) \cdot ξ - B$ of The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate and the energy charge $- E$ of Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential as special cases; the framework's time-translation instance uses a parameter measured in seconds, so $τ \equiv 1$ is dimensionless and $- E$ is action per second. The conservation laws are therefore instances of one identity rather than independent postulates. The Lorentz-type generator with both $τ$ and $ξ$ nonzero — for which the charge mixes position, time and momentum — is treated as scope, not as a force law, and no such model is asserted here.
Sign convention. The convention is the one fixed by Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions: $ξ$ generates the configuration change $x \mapsto x + s ξ$ at fixed $t$ up to the time shift $s τ$ . Reversing the family parameter negates the generator, boundary function and charge together, preserving the identity and conservation statement. The printed rotation derivative in SICM Section 1.8.5 has the sign inconsistency identified in the symmetry definition; it is not used here.
Conditional conservation and empirical status. Clause 3 is the exact bookkeeping for a presentation with residual forces: the residual force acts against the total generator $ξ - x^{'} τ$ , so a symmetry of the potential part produces a conserved charge only when the residual force pairs to zero with that generator along the motion. The identity of clause (a) of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions must be verified for the actual declared $U$ and masses; the conservation statements are deductions inside the model, not empirical laws, and no converse is asserted.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential

Physical scope

Time translation as a symmetry datum of the force model with a declared potential presentation: with a time-valued family parameter in seconds, the datum (tau,xi,B)=(1,0,0) satisfies the model's Noether identity exactly when the potential has no explicit time dependence; in the global potential branch with an autonomous potential the mechanical energy K+U is constant along every motion; and in that branch, if the potential depends explicitly on time at a state, then the maximal motion through that state has nonvanishing energy derivative there, so the energy is not conserved. The statement is conditional on the adopted force branch and the declared presentation; the sharpness clause uses the maximal-motion existence theorem, and no empirical content is asserted.

Statement

$E (t, x, v) : = K (v) + U (t, x) = \frac{1}{2} v \cdot M v + U (t, x)$

be the model's mechanical energy function. Consider the datum $τ \equiv 1$ , $ξ \equiv 0$ , $B \equiv 0$ of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions. Then:

For this time-translation family take its parameter $s$ to be measured in seconds; $τ \equiv 1$ is dimensionless and the charge $- E$ has units of action per second, namely joules.

(the identity). The datum satisfies the model's Noether identity on $O$ if and only if $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ ; its charge is the function $- E$ .
(conservation). If $d \equiv 0$ and $\partial_{t} U \equiv 0$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto E (t, x (t), x^{'} (t)) = K (x^{'} (t)) + U (t, x (t))$ is constant on $J$ .
(sharpness). If $d \equiv 0$ and there is a state $(t_{0}, x_{0}) \in I_{0} \times Q$ with $\partial_{t} U (t_{0}, x_{0}) \neq 0$ , then the maximal motion $x_{\max} : J_{\max} \to Q$ of the model through the initial datum $(t_{0}, x_{0}, 0)$ satisfies $\frac{d}{d t} (K (x_{\max}^{}) + U (\cdot, x_{\max})) ∣_{t_{0}} = \partial_{t} U (t_{0}, x_{0}) \neq 0$ , and the mechanical energy is not constant on $J_{\max}$ .

The conservation clause is the energy case of the model's Noether identity and is conditional on the adopted force branch, the declared representative $U$ and the absence of residual forces; the sharpness clause shows that explicit time dependence of the declared representative destroys conservation along an actual motion. No empirical content is asserted.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the mechanical energy function $E$ , and the motions and state of the clauses.

[F1]

Model data, chart and charge dictionary (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $O = I_{0} \times Q \times R^{3 N}$ is open and $(O, L)$ is a time-dependent Lagrangian chart whenever $Q \neq \emptyset$ and $a < b$ with $[a, b] \subseteq I_{0}$ , over $[a, b] \times Q$ ; $L = K - U$ with $K (v) = \frac{1}{2} v \cdot M v$ independent of $t$ and $x$ , so $L_{t} = - U_{t}$ and the mechanical energy function is $E = L_{v} \cdot v - L = K + U$ ; the charge of a datum $(τ, ξ, B)$ is $(M v) \cdot ξ - E τ - B$ .

[F2]

The general Noether theorem of the model (The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p \cdot ξ - E τ - B$ , its residual-force rate and its potential gauge): if a symmetry datum satisfies the model identity on $O$ and $d \equiv 0$ , then its charge is constant along every motion; at $ξ \equiv 0$ , $τ \equiv 1$ , $B \equiv 0$ the model identity holds on $O$ if and only if $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ , and the charge is $- E$ .

[F3]

Energy balance of the model (Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form clause 1, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance): along a motion of the presentation $(U, d)$ the mechanical energy is differentiable with $E^{'} (t) = \partial_{t} U (t, x (t)) + \sum_{i} ⟨ d_{i} (t, x (t), x^{'} (t)), x_{i}^{'} (t) ⟩$ ; with $d \equiv 0$ this reduces to $E^{'} (t) = \partial_{t} U (t, x (t))$ .

[F4]

Motions and maximal motions (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): a motion $x : J \to Q$ is $C^{2}$ with $J \subseteq I_{0}$ and $M x^{''} = - \nabla_{x} U + d$ on $J$ ; through every initial datum $(t_{0}, x_{0}, v_{0})$ there is exactly one maximal motion, its maximal interval is an open interval containing $t_{0}$ and contained in $I_{0}$ , and every motion with the same datum is its restriction.

[F5]

Calculus and dimensions (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): $E$ is of the stated regularity along motions; a $C^{1}$ function on the interval $J_{\max}$ whose derivative vanishes everywhere is constant, and a function that is constant on $J_{\max}$ has zero derivative at every point of $J_{\max}$ ; all quantities carry their declared dimensions.

Proof

technique · direct; evaluate the model identity at the time-translation datum, transfer the conservation clause through the model's general Noether theorem and the sharpness clause through the energy balance along the maximal motion

1.1F1F2algebra

Clause 1, evaluation of the identity: at the datum $τ \equiv 1$ , $ξ \equiv 0$ , $B \equiv 0$ the left-hand side of the model identity of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions clause (a) is $L_{t} \cdot 1 + 0 + 0 + L \cdot 0 = L_{t}$ and the right-hand side is $D_{t} B = 0$ , so the identity holds on $O$ if and only if $L_{t} \equiv 0$ ; by [F1], $L_{t} = - U_{t}$ , so this is equivalent to $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ , and the charge is $(M v) \cdot 0 - E \cdot 1 - 0 = - E$ . This is exactly the specialisation recorded in clause 2 of [F2].

1.2F3F4given

Clause 3, the derivative of the energy: let $(t_{0}, x_{0}) \in I_{0} \times Q$ and let $x_{\max} : J_{\max} \to Q$ be the maximal motion through $(t_{0}, x_{0}, 0)$ , which exists and has open $J_{\max} ∋ t_{0}$ by [F4]; since $d \equiv 0$ , [F3] gives that the function $t \mapsto E (t, x_{\max} (t), x_{\max}^{} (t))$ is differentiable on $J_{\max}$ with derivative $\partial_{t} U (t, x_{\max} (t))$ at every $t \in J_{\max}$ , in particular at $t_{0}$ its derivative is $\partial_{t} U (t_{0}, x_{0})$ .

2.1F2step 1.1given

Clause 2, conservation: assume $d \equiv 0$ and $\partial_{t} U \equiv 0$ ; by step 1.1 the datum satisfies the model identity on $O$ , so clause 1 of [F2] applies to every motion and gives that the charge $- E$ is constant along it; hence $E$ is constant on $J$ .

2.2F5step 1.2

Clause 3, non-constancy: with $d \equiv 0$ and $\partial_{t} U (t_{0}, x_{0}) \neq 0$ , step 1.2 gives that the derivative of the mechanical energy along $x_{\max}$ at $t_{0}$ equals $\partial_{t} U (t_{0}, x_{0}) \neq 0$ ; were the energy constant on $J_{\max}$ , its derivative would vanish at every point of $J_{\max}$ by [F5], a contradiction.

3.1step 1.1step 2.1step 2.2∎

Clause 1 is step 1.1, clause 2 is step 2.1 and clause 3 is steps 1.2 and 2.2; the conservation clause is the energy case of the model's Noether identity, the sharpness clause exhibits the failure of conservation along an actual motion, and no existence of additional symmetry data is asserted.

Remarks

Why the two clauses are sharp together. Clause 1 is an equivalence at the level of the identity: the datum satisfies the model identity exactly for autonomous representatives of the potential. Clause 3 shows that a time-dependent representative does not merely fail to supply the Noether hypothesis: along a motion through a state where the explicit time dependence is nonzero, the energy derivative is nonzero there, so the failure is observable inside the model. This is a statement about the declared representative; the representative is part of the presentation (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)) and is named in every energy statement.
Residual forces. For a presentation with $d \equiv̸ 0$ the rate formula of Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form clause 1 applies verbatim; the energy is then not conserved in general even for an autonomous potential, and the general Noether theorem of this page records how a residual force enters the charge rates of symmetry data. No separate energy identity is claimed here beyond the cited balance.
Empirical status. The word "energy" here denotes the mechanical energy function $K + U$ of the declared presentation, a model object. Its conservation is a deduction inside the model under the stated hypotheses; it is not a report of a measurement and does not by itself assert that any physical system has an autonomous potential presentation.

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Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness

Physical scope

Translation symmetry of the force model with a declared potential presentation in one inertial frame: with a dimensionless direction vector and a length-valued translation parameter, the exact equivalence between the differential invariance condition sum_i a . grad_i U = 0 and finite invariance on admissible translation segments, with the exact component compatibility criterion for all admissible endpoints, the resulting strict symmetry datum and conservation of the component a . P of the total momentum along every motion of the global potential branch, the sharpness statement in that branch that a nonzero directional derivative at a state produces a motion whose momentum component is not constant, and the vector form for a fully translation-invariant potential. The theorem is conditional on the adopted force branch, the declared potential and the absence of residual forces in the conservation clauses; it asserts no translation symmetry of any particular force law and carries no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. For the translation family take the parameter $s$ to have units of length and let $a \in R^{3}$ be a nonzero dimensionless vector. Let $ξ^{a} \in R^{3 N}$ be the configuration vector whose $i$ -th three-dimensional block is $ξ_{i}^{a} : = a$ , so that $(M v) \cdot ξ^{a} = \sum_{i} m_{i} a \cdot v_{i}$ for a velocity $v$ . Write $P (t) : = \sum_{i} m_{i} x_{i}^{'} (t)$ for the total momentum of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame. Then:

(infinitesimal and finite invariance). The following are equivalent: (i) $\sum_{i = 1}^{N} a \cdot \nabla_{x_{i}} U (t, x) = 0$ for every $(t, x) \in I_{0} \times Q$ ; (ii) $U (t, x + s ξ^{a}) = U (t, x)$ whenever the whole segment ${x + r ξ^{a} : r between 0 and s}$ lies in $Q$ . More generally, for fixed $(t, x)$ put $J_{x} : = {s \in R : x + s ξ^{a} \in Q}$ , an open set containing $0$ . Its interval components are its maximal open intervals. Under (i), $s \mapsto U (t, x + s ξ^{a})$ is constant on each interval component. Invariance for every $s \in J_{x}$ , including components separated by an excluded segment, holds if and only if all these component constants equal $U (t, x)$ . This agreement is automatic when $J_{x}$ is connected, in particular when the whole translation line lies in $Q$ .
(the symmetry datum). Under these equivalent hypotheses the strict datum $τ \equiv 0$ , $ξ \equiv ξ^{a}$ , $B \equiv 0$ satisfies the model's Noether identity on $O$ in the sense of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions.
(conservation of the momentum component). If in addition $d \equiv 0$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto a \cdot P (t) = \sum_{i} ⟨ a, m_{i} x_{i}^{'} (t) ⟩$ is constant on $J$ .
(sharpness). If $d \equiv 0$ and there is a state $(t_{0}, x_{0}) \in I_{0} \times Q$ with $g : = \sum_{i} a \cdot \nabla_{x_{i}} U (t_{0}, x_{0}) \neq 0$ , then the maximal motion $x_{\max} : J_{\max} \to Q$ through the initial datum $(t_{0}, x_{0}, 0)$ satisfies $\frac{d}{d t} (a \cdot P (t)) ∣_{t_{0}} = - g \neq 0$ ; in particular $a \cdot P$ is not constant on $J_{\max}$ .
(vector form). If $\sum_{i} \nabla_{x_{i}} U (t, x) = 0$ for every $(t, x) \in I_{0} \times Q$ , then with $d \equiv 0$ the total momentum vector $P$ is constant along every motion.

The hypotheses are conditions on the declared potential; the conservation clauses are conditional deductions of the global potential branch, the sharpness clause locates the failure at an actual motion, and no symmetry of any particular force law and no empirical content is asserted.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the vector $a$ , the datum $ξ^{a}$ , and the motions and state occurring in the clauses.

[F1]

Model data and chart (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $O = I_{0} \times Q \times R^{3 N}$ is open and $(O, L)$ is a time-dependent Lagrangian chart whenever $Q \neq \emptyset$ and $a^{'} < b^{'}$ with $[a^{'}, b^{'}] \subseteq I_{0}$ , over $[a^{'}, b^{'}] \times Q$ ; $L = K - U$ with $L_{x} (t, x, v) = - \nabla_{x} U (t, x)$ , $L_{v} (t, x, v) = M v$ ; the force map is $F_{i} = - \nabla_{x_{i}} U + d_{i}$ .

[F2]

The model's symmetry vocabulary (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): a strict datum $τ \equiv 0, ξ, B$ satisfies the model identity when $L_{x} \cdot ξ + L_{v} \cdot D_{t} ξ = D_{t} B$ on $O$ , and its charge is $(M v) \cdot ξ - B$ ; a strict datum with $ξ$ independent of $t$ is an infinitesimal point symmetry of the chart $(O, L)$ .

[F3]

The point Noether theorem of the model (The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate clause 1): if the strict datum has $ξ$ independent of $t$ , satisfies the model identity and $d \equiv 0$ , then its charge is constant along every motion.

[F4]

Motions and maximal motions (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance): a motion $x : J \to Q$ satisfies $M x^{''} = - \nabla_{x} U + d$ on $J$ ; through every initial datum there is exactly one maximal motion with open maximal interval containing the initial time.

[F5]

Momentum quantities (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame): $P = \sum_{i} m_{i} x_{i}^{'}$ is the total momentum, a curve in $R^{3}$ when the $x_{i}$ are $C^{1}$ .

[F6]

Calculus along the motion and translation family (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ): $J_{x}$ is an open set containing $0$ ; each of its interval components is a maximal open interval, and on it $s \mapsto U (t, x + s ξ^{a})$ has derivative $\sum_{i} a \cdot \nabla_{x_{i}} U (t, x + s ξ^{a})$ . A $C^{1}$ function on an interval with zero derivative is constant; openness alone does not connect distinct components.

[F7]

Parameter units (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): the translation parameter has units of length and $a$ is dimensionless, so the charge $a \cdot P$ has dimension $μ ℓ τ^{- 1}$ , equal to action per unit length.

Proof

technique · direct; prove the equivalence of the two forms of the hypothesis by differentiating and integrating along the translation segment, verify the strict identity, transfer conservation from the model's point Noether theorem, and compute the failure rate along the maximal motion

1.1F6givenalgebra

Clause 1, (i) implies (ii): for fixed $(t, x)$ the set $J_{x}$ is open by continuity of $s \mapsto x + s ξ^{a}$ , but can have several interval components. On each such interval $f (s) : = U (t, x + s ξ^{a})$ is $C^{1}$ with derivative $\sum_{i} a \cdot \nabla_{x_{i}} U (t, x + s ξ^{a}) = 0$ under (i), hence is constant by [F6]. If the entire segment between $0$ and $s$ is admissible, both parameters lie in the same interval component, so $f (s) = f (0)$ , proving (ii). The same calculation proves constancy on every component; equality at every admissible endpoint holds exactly when all component constants equal $f (0)$ , and is automatic for connected $J_{x}$ .

1.2F1F2givenalgebra

Clause 2, the identity: for the datum $τ \equiv 0$ , $ξ \equiv ξ^{a}$ , $B \equiv 0$ one has $D_{t} ξ^{a} = 0$ because $ξ^{a}$ is constant, and $L_{x} \cdot ξ^{a} = - \sum_{i} a \cdot \nabla_{x_{i}} U$ ; under hypothesis (i) the left-hand side of the strict identity is therefore $- \sum_{i} a \cdot \nabla_{x_{i}} U + (M v) \cdot 0 = 0$ , while the right-hand side is $D_{t} B = 0$ ; so the model identity holds on $O$ .

1.3F1F4F5algebra

Clause 4, the rate: along a motion with $d \equiv 0$ , the function $t \mapsto a \cdot P (t) = \sum_{i} ⟨ a, m_{i} x_{i}^{'} (t) ⟩$ is differentiable by [F4] and [F5] with $\frac{d}{d t} (a \cdot P) = \sum_{i} ⟨ a, m_{i} x_{i}^{''} (t) ⟩ = - \sum_{i} a \cdot \nabla_{x_{i}} U (t, x (t));$ hence at the state $(t_{0}, x_{0})$ the maximal motion through $(t_{0}, x_{0}, 0)$ has derivative $- g$ there.

2.1F6step 1.1algebra

Clause 1, (ii) implies (i): if (ii) holds then for each $(t, x)$ the function $f$ of step 1.1 is constant on a neighbourhood of $0$ in $J_{x}$ , so its derivative at $0$ , namely $\sum_{i} a \cdot \nabla_{x_{i}} U (t, x)$ , vanishes; this is (i).

2.2F2F3step 1.2

Clause 3, conservation: under hypothesis (i) and $d \equiv 0$ , step 1.2 exhibits a strict datum with $ξ^{a}$ independent of $t$ satisfying the model identity, so clause 1 of [F3] applies to every motion and gives that its charge $(M x^{'}) \cdot ξ^{a} - 0 = \sum_{i} ⟨ a, m_{i} x_{i}^{'} ⟩ = a \cdot P$ is constant on $J$ .

2.3F6step 1.3

Clause 4, non-constancy: with $d \equiv 0$ and $g \neq 0$ , step 1.3 gives that the derivative of $a \cdot P$ along the maximal motion at $t_{0}$ is $- g \neq 0$ ; a function constant on the interval $J_{\max}$ would have vanishing derivative there by [F6].

3.1F5step 2.2algebra

Clause 5: assume $\sum_{i} \nabla_{x_{i}} U = 0$ and $d \equiv 0$ ; then hypothesis (i) holds for each of the three coordinate vectors $a = e_{1}, e_{2}, e_{3}$ , so step 2.2 gives that each component $e_{k} \cdot P$ of the total momentum is constant along every motion, which is the assertion that the vector $P$ is constant.

4.1step 1.1step 1.2step 2.2step 2.3step 3.1F7∎

Clause 1 is steps 1.1 and 2.1, clause 2 is step 1.2, clause 3 is step 2.2, clause 4 is steps 1.3 and 2.3, and clause 5 is step 3.1; the conservation clauses are conditional on $d \equiv 0$ and the stated invariance of the declared potential, and no converse from conservation to a symmetry and no empirical claim is made. By [F7], $[a \cdot P] = μ ℓ τ^{- 1}$ , the action dimension divided by the length dimension of the translation parameter.

Remarks

The shape of the hypothesis. Condition (i) is infinitesimal translation invariance and needs no convexity or connectedness of $Q$ . It is equivalent to finite invariance along every segment contained in $Q$ . Comparing admissible endpoints across an excluded segment additionally requires the component constants of clause 1 to agree. This qualification cannot be omitted: on two disjoint balls with centres on the translation axis, a potential equal to $0$ on one and $1$ on the other has zero directional derivative everywhere but fails endpoint invariance. The momentum conservation and sharpness clauses retain their full scope under the differential hypothesis on arbitrary open $Q$ . For a pair-presented translation-invariant potential that hypothesis holds identically; the Noether route uses the potential presentation and $d \equiv 0$ , while Total momentum balance and centre-of-mass equation for a model with antisymmetric pair forces uses antisymmetric pair forces, and neither route implies the other's hypotheses.
Frame dependence of the charge. The conserved quantity is the component of the total momentum in the fixed inertial frame, with the generator frozen at the vector $a$ ; under a change of inertial frame the description of the model changes and the charge transforms with the transformation of $P$ and of the transformed generator, so the conservation statement is frame-relative in the sense that it is proved in one inertial frame. No relativistic interpretation of the charge is intended.
Empirical status. The statement identifies a conserved quantity of the model under an explicit symmetry hypothesis of the declared potential; it is a conditional deduction within the model and carries no measurement, no claim that any actual force law is translation invariant, and no converse.

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Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin

Physical scope

Rotation symmetry of the force model with a declared potential presentation in one inertial frame: with a dimensionless generator vector and a rotation parameter measured in radians, the equivalence between the differential invariance condition of the potential along the rotation family with axis a and finite invariance on admissible rotation arcs, with the exact component compatibility criterion for all admissible endpoints, the resulting strict symmetry datum, conservation of the component a . J of the total angular momentum about the inertial origin along every motion of the global potential branch, its sharpness in that branch under a nonzero rotational derivative of U, and the vector form when the potential is invariant under all rotations about the origin. The theorem is conditional on the adopted force branch, the declared potential and the absence of residual forces in the conservation clauses; it asserts no rotation symmetry of any particular force law and carries no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. Take the rotation parameter $s$ to be measured in radians, treated as dimensionless in SI, and take $a$ to be dimensionless; thus $∣ a ∣ s$ is dimensionless and the charge $a \cdot J$ has angular-momentum units (action). Fix a nonzero vector $a \in R^{3}$ and put, for $s \in R$ and $z \in R^{3}$ ,

$R_{a} (s) z : = z + \frac{\sin (∣ a ∣ s)}{∣ a ∣} a \times z + \frac{1 - \cos (∣ a ∣ s)}{∣ a ∣^{2}} a \times (a \times z),$

where $\times$ is the cross product of $R^{3}$ (The cross product in $R^{3}$ ). For a configuration $x = (x_{1}, \dots, x_{N})$ write $R_{a} (s) x : = (R_{a} (s) x_{1}, \dots, R_{a} (s) x_{N})$ , and let $ξ^{a} : Q \to R^{3 N}$ be the vector field with $ξ_{i}^{a} (x) : = a \times x_{i}$ . Write $J (t) : = \sum_{i} x_{i} (t) \times m_{i} x_{i}^{'} (t)$ for the total angular momentum about the inertial origin (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame). Then:

(infinitesimal and finite invariance). The following are equivalent: (i) $\sum_{i} ⟨ a \times x_{i}, \nabla_{x_{i}} U (t, x) ⟩ = 0$ for every $(t, x) \in I_{0} \times Q$ ; (ii) $U (t, R_{a} (s) x) = U (t, x)$ whenever ${R_{a} (r) x : r between 0 and s}$ lies in $Q$ . More generally put $D_{x} : = {s \in R : R_{a} (s) x \in Q}$ , an open set containing $0$ , with interval components its maximal open intervals. Under (i), $s \mapsto U (t, R_{a} (s) x)$ is constant on each interval component. Invariance for every $s \in D_{x}$ , including components separated by an excluded arc, holds if and only if all these constants equal $U (t, x)$ . This agreement is automatic if $D_{x}$ is connected, in particular if the whole rotation orbit is contained in $Q$ .
(the symmetry datum). Under these equivalent hypotheses the strict datum $τ \equiv 0$ , $ξ \equiv ξ^{a}$ , $B \equiv 0$ satisfies the model's Noether identity on $O$ in the sense of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions.
(conservation of the angular-momentum component). If in addition $d \equiv 0$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto a \cdot J (t) = \sum_{i} ⟨ a, x_{i} (t) \times m_{i} x_{i}^{'} (t) ⟩$ is constant on $J$ .
(sharpness). If $d \equiv 0$ and there is a state $(t_{0}, x_{0}) \in I_{0} \times Q$ with $g : = \sum_{i} ⟨ a \times x_{0, i}, \nabla_{x_{i}} U (t_{0}, x_{0}) ⟩ \neq 0$ , then the maximal motion $x_{\max} : J_{\max} \to Q$ through the initial datum $(t_{0}, x_{0}, 0)$ satisfies $\frac{d}{d t} (a \cdot J (t)) ∣_{t_{0}} = - g \neq 0$ ; in particular $a \cdot J$ is not constant on $J_{\max}$ .
(vector form). If $\sum_{i} x_{i} \times \nabla_{x_{i}} U (t, x) = 0$ for every $(t, x) \in I_{0} \times Q$ , then with $d \equiv 0$ the total angular momentum vector $J$ is constant along every motion.

The hypothesis is a condition on the declared potential — equivalently, by clause 1, invariance of $U$ along admissible arcs of the simultaneous rotations $R_{a} (s)$ of all labels, with full endpoint invariance requiring the component agreement specified there; the conservation clauses are conditional deductions of the global potential branch, and no symmetry of any particular force law and no empirical content is asserted.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the vector $a$ , the family $R_{a}$ , the field $ξ^{a}$ , and the motions and state occurring in the clauses.

[F1]

[F2]

The model's symmetry vocabulary (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): a strict datum $τ \equiv 0, ξ, B$ satisfies the model identity when $L_{x} \cdot ξ + L_{v} \cdot D_{t} ξ = D_{t} B$ on $O$ , with charge $(M v) \cdot ξ - B$ ; a strict datum with $ξ$ independent of $t$ is an infinitesimal point symmetry of the chart $(O, L)$ .

[F3]

The point Noether theorem of the model (The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate clause 1): a strict datum with $ξ$ independent of $t$ satisfying the model identity and $d \equiv 0$ has constant charge along every motion.

[F4]

The cross product (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors): the product is bilinear and alternating, $u \times u = 0$ , $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to $u$ and $v$ ; in particular $⟨ u, u \times w ⟩ = 0$ and $⟨ a, u \times w ⟩ = ⟨ a \times u, w ⟩$ for all $a, u, w$ .

[F5]

Trigonometry and calculus of the family (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, $C^{k}$ Euclidean maps and diffeomorphisms, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): sine and cosine are differentiable with the displayed derivatives, $\sin 0 = 0$ , $\cos 0 = 1$ and $\sin^{2} + \cos^{2} = 1$ ; sums and scalar multiples of differentiable curves differentiate by the usual rules.

[F6]

Calculus along the motion and rotation family (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): $D_{x}$ is open and contains $0$ ; its interval components are maximal open intervals. On each such interval $s \mapsto U (t, R_{a} (s) x)$ is $C^{1}$ with derivative $\sum_{i} ⟨ \nabla_{x_{i}} U (t, R_{a} (s) x), \frac{d}{d s} R_{a} (s) x_{i} ⟩$ . A $C^{1}$ function on an interval with zero derivative is constant; openness alone does not join separated admissible arcs.

[F7]

Moment and motion identities (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance): $J = \sum_{i} x_{i} \times m_{i} x_{i}^{'}$ with $m_{i} x_{i}^{''} = - \nabla_{x_{i}} U + d_{i}$ along a motion; through every initial datum there is exactly one maximal motion with open maximal interval containing the initial time.

Proof

technique · direct; compute the derivative of the rotation family and identify the rotational derivative of the potential, verify the strict Noether identity, transfer conservation from the model's point Noether theorem and compute the failure rate

1.1F4F5algebra

The family and its derivative: for all $s \in R$ and $z \in R^{3}$ one has $R_{a} (s) a = a$ , $∣ R_{a} (s) z ∣ = ∣ z ∣$ , and $\frac{d}{d s} (R_{a} (s) z) = a \times (R_{a} (s) z) .$ Indeed, expanding the three components of $a \times (a \times z)$ with the coordinate formula of [F4] gives the identity $a \times (a \times z) = a (a \cdot z) - ∣ a ∣^{2} z$ , whence $a \times (a \times (a \times z)) = - ∣ a ∣^{2} (a \times z)$ ; differentiating the displayed definition by [F5] gives $\frac{d}{d s} (R_{a} (s) z) = \cos (∣ a ∣ s) (a \times z) + \frac{\sin (∣ a ∣ s)}{∣ a ∣} a \times (a \times z)$ , and substituting the two identities into $a \times (R_{a} (s) z)$ gives the same expression; the first two claims follow from $a \times a = 0$ and from $\frac{d}{d s} ∣ R_{a} (s) z ∣^{2} = 2 ⟨ R_{a} (s) z, a \times R_{a} (s) z ⟩ = 0$ together with $∣ R_{a} (0) z ∣ = ∣ z ∣$ .

1.2F1F2F4givenalgebra

Clause 2, the identity: for $ξ = ξ^{a}$ one has $D ξ^{a} (x) v = (a \times v_{1}, \dots, a \times v_{N})$ because $x_{i} \mapsto a \times x_{i}$ is linear, so the strict identity reads $L_{x} \cdot ξ^{a} + L_{v} \cdot D ξ^{a} v = - \sum_{i} ⟨ a \times x_{i}, \nabla_{x_{i}} U ⟩ + \sum_{i} ⟨ m_{i} v_{i}, a \times v_{i} ⟩;$ the second sum vanishes because $⟨ v_{i}, a \times v_{i} ⟩ = 0$ by [F4], and the first vanishes under hypothesis (i); hence the left-hand side is $0 = D_{t} B$ .

1.3F4F7algebra

Clause 4, the rate: along a motion with $d \equiv 0$ , the function $t \mapsto a \cdot J (t)$ is differentiable with $\frac{d}{d t} (a \cdot J) = \sum_{i} ⟨ a, x_{i}^{'} \times m_{i} x_{i}^{'} + x_{i} \times m_{i} x_{i}^{''} ⟩ = \sum_{i} ⟨ a, x_{i} \times (- \nabla_{x_{i}} U) ⟩ = - \sum_{i} ⟨ a \times x_{i}, \nabla_{x_{i}} U ⟩,$ where the second equality uses $x_{i}^{'} \times x_{i}^{'} = 0$ and $m_{i} x_{i}^{''} = - \nabla_{x_{i}} U$ , and the last uses $⟨ a, x_{i} \times \nabla_{x_{i}} U ⟩ = ⟨ a \times x_{i}, \nabla_{x_{i}} U ⟩$ ; at the state $(t_{0}, x_{0})$ the maximal motion through $(t_{0}, x_{0}, 0)$ therefore has derivative $- g$ there.

2.1F6step 1.1given

Clause 1, (i) implies (ii): $D_{x}$ is open by continuity of the rotation family, but can have several interval components. On each component $f (s) : = U (t, R_{a} (s) x)$ is $C^{1}$ and has derivative $\sum_{i} ⟨ \nabla_{x_{i}} U (t, R_{a} (s) x), a \times R_{a} (s) x_{i} ⟩ = 0$ by [F6], step 1.1 and (i). It is therefore constant there. When the whole arc between $0$ and $s$ lies in $Q$ , both parameters lie in the same interval component and $f (s) = f (0)$ , which is (ii). The calculation also gives constancy on every component; equality at every admissible endpoint is equivalent to agreement of all component constants with $f (0)$ , automatic for connected $D_{x}$ .

2.2F2F3step 1.2algebra

Clause 3, conservation: under hypothesis (i) and $d \equiv 0$ , step 1.2 exhibits a strict datum with $ξ^{a}$ independent of $t$ satisfying the model identity, so clause 1 of [F3] applies to every motion and gives that the charge $(M x^{'}) \cdot ξ^{a} = \sum_{i} ⟨ m_{i} x_{i}^{'}, a \times x_{i} ⟩ = \sum_{i} ⟨ a, x_{i} \times m_{i} x_{i}^{'} ⟩ = a \cdot J$ is constant on $J$ .

2.3F6step 1.3

Clause 4, non-constancy: with $d \equiv 0$ and $g \neq 0$ , step 1.3 gives that the derivative of $a \cdot J$ along the maximal motion at $t_{0}$ is $- g \neq 0$ ; a function constant on the interval $J_{\max}$ would have vanishing derivative there by [F6].

3.1F6step 1.1step 2.1

Clause 1, (ii) implies (i): if (ii) holds then for each $(t, x)$ the function $f$ of step 2.1 is constant on a neighbourhood of $0$ , so its derivative at $0$ vanishes; by [F6] and step 1.1 that derivative is $\sum_{i} ⟨ \nabla_{x_{i}} U (t, x), a \times x_{i} ⟩$ , which is (i).

3.2F4step 2.2algebra

Clause 5: assume $\sum_{i} x_{i} \times \nabla_{x_{i}} U = 0$ and $d \equiv 0$ ; by [F4], $\sum_{i} ⟨ a \times x_{i}, \nabla_{x_{i}} U ⟩ = ⟨ a, \sum_{i} x_{i} \times \nabla_{x_{i}} U ⟩ = 0$ for every $a \in R^{3}$ , so hypothesis (i) holds for each of the coordinate vectors $a = e_{1}, e_{2}, e_{3}$ , and step 2.2 gives that each component $e_{k} \cdot J$ of the angular momentum is constant along every motion, which is the assertion that the vector $J$ is constant.

4.1step 1.1step 1.2step 2.2step 2.3step 3.2∎

Clause 1 is steps 2.1 and 3.1, clause 2 is step 1.2, clause 3 is step 2.2, clause 4 is steps 1.3 and 2.3, and clause 5 is step 3.2; the conservation clauses are conditional on $d \equiv 0$ and the rotational invariance of the declared potential, and no converse from conservation to a symmetry and no empirical claim is made.

Remarks

Separated arcs. On two disjoint balls centred at opposite points of a circle about the axis, the potential equal to $0$ on one ball and $1$ on the other has zero rotational derivative everywhere, but a rotation between the two centres changes its value. Differential conservation holds on arbitrary open $Q$ ; comparing separated arcs requires the compatibility in clause 1.
Torque reading. Under the potential presentation the total torque of the potential forces about the origin is $- \sum_{i} x_{i} \times \nabla_{x_{i}} U$ , so hypothesis (i) for all $a$ is the vanishing of that torque; clause 5 is then the familiar statement that a model whose potential exerts no torque about the origin conserves its total angular momentum. The balance theorem Angular momentum balance about a fixed or moving reference point, the internal torque sum, and the centrality case reaches the conservation statement from centrality of the pair forces and vanishing external torque, a different hypothesis; the two routes are complementary and neither is a consequence of the other.
Angular momentum about other origins. The charge constructed here is the angular momentum about the inertial origin, which is where the rotation family $R_{a}$ acts. For rotations about another fixed point $o$ the same computation with $x_{i} - o$ in place of $x_{i}$ produces the component of the angular momentum about $o$ ; that variant is not needed on this page and is not asserted.
Empirical status. The statement identifies a conserved quantity of the model under an explicit rotational invariance hypothesis of the declared potential; it is a conditional deduction inside the model, and no claim is made that any actual force law is rotation invariant, nor is any converse from an observed conservation to a symmetry of a model.

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The Galilean boost charge of the model: exact condition, uniform centre-of-mass motion, the induced motions and the boundary of the conservation law

Physical scope

The Galilean boost charge of the force model with a declared potential presentation in one inertial frame: with dimensionless a and a velocity-valued family parameter, the time-dependent triple (tau, xi, B) = (0, t a, a . M x) satisfies the model's Noether identity exactly when the differential translation condition sum_i a . grad_i U = 0 holds; in the global potential branch the boost charge a . (t P - M X) is then constant, the component a of the centre of mass moves uniformly, and admissible boosted curves t -> x(t) + s t a are again motions precisely under the endpoint-gradient agreement in Statement clause 4; for an arbitrary presentation the exact rate of the charge is the time times the component a of the total force, while nonconservation from a nonzero translation derivative is asserted in the global potential branch. The charge has dimension mass times length (action per velocity), and its rate has momentum dimensions. The theorem is conditional on the adopted force branch, the declared potential and the absence of residual forces in the conservation clauses; it asserts no boost invariance of any particular force law and carries no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ and $U$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets. For this boost family take the parameter $s$ to have units of velocity and take $a \in R^{3}$ to be dimensionless; then $s t a$ is a displacement, $ξ^{a} = t a$ with $a : = (a, \dots, a) \in R^{3 N}$ has units of time, and $B^{a}$ and $K_{a}$ have units $k g m$ , equal to action per unit velocity. Fix a nonzero vector $a \in R^{3}$ and consider the maps

$ξ^{a} (t, x) : = t a, B^{a} (t, x) : = a \cdot \sum_{i} m_{i} x_{i},$

These maps, together with $τ \equiv 0$ , form a $C^{2}$ triple on $I_{0} \times Q$ . It is a symmetry datum in the sense of Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions exactly when it satisfies the Noether identity, as characterised in clause 1 below. Write $P (t) : = \sum_{i} m_{i} x_{i}^{'} (t)$ for the total momentum and $X (t)$ for the centre of mass of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, where the scalar total mass is $M : = \sum_{i} m_{i}$ and $M X (t) = \sum_{i} m_{i} x_{i} (t)$ ; thus along a motion $B^{a} (t, x (t)) = M a \cdot X (t)$ . Define the boost charge

$K_{a} (t) : = a \cdot (t P (t) - M X (t)) = t (a \cdot P (t)) - \sum_{i} m_{i} a \cdot x_{i} (t) .$

Then:

(the identity). The triple $(0, ξ^{a}, B^{a})$ satisfies the model's Noether identity on $O$ if and only if the translation-invariance condition $\sum_{i} a \cdot \nabla_{x_{i}} U (t, x) = 0$ for all $(t, x) \in I_{0} \times Q$ of clause 1 of Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness holds.
(conservation). If in addition $d \equiv 0$ , then for every motion $x : J \to Q$ of the model on an interval $J$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model), the function $t \mapsto K_{a} (t)$ is constant on $J$ .
(uniform centre-of-mass motion). Under the hypotheses of clause 2, the function $t \mapsto M a \cdot X (t)$ is affine on $J$ with slope $a \cdot P$ .
(boosted motions). Under the hypotheses of clause 2, let $x : J \to Q$ be a motion and $s \in R$ satisfy $x (t) + s t a \in Q$ for every $t \in J$ . Then $\tilde{x} (t) : = x (t) + s t a$ is a motion if and only if $\nabla_{x} U (t, \tilde{x} (t)) = \nabla_{x} U (t, x (t))$ for every $t \in J$ . This equality, and hence the boosted-motion conclusion, follows if each segment between $x (t)$ and $\tilde{x} (t)$ is contained in $Q$ , or if the all-admissible-endpoint finite invariance criterion of Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness clause 1 holds.
(the general rate and the boundary). For an arbitrary presentation $(U, d)$ and every motion $x : J \to Q$ ,

$K_{a}^{'} (t) = t a \cdot F_{t o t} (t), F_{t o t} (t) : = \sum_{i} F_{i} (t, x (t), x^{'} (t));$

consequently, when $d \equiv 0$ , the charge $K_{a}$ is constant on $J$ if and only if $t \mapsto t \sum_{i} a \cdot \nabla_{x_{i}} U (t, x (t))$ vanishes identically on $J$ , and if the translation-invariance condition fails at some state, then there are a time $t_{0} \neq 0$ and a state $x_{0}$ with $g : = \sum_{i} a \cdot \nabla_{x_{i}} U (t_{0}, x_{0}) \neq 0$ such that the maximal motion through $(t_{0}, x_{0}, 0)$ satisfies $K_{a}^{'} (t_{0}) = - t_{0} g \neq 0$ and $K_{a}$ is not constant on its interval.

The charge exists as a conserved quantity exactly under the same potential invariance that conserves the momentum component $a \cdot P$ ; the conservation clauses are conditional deductions of the global potential branch, and no empirical content is asserted.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the vector $a$ , the triple $(0, ξ^{a}, B^{a})$ , and the motions and states occurring in the clauses.

[F1]

Model data and chart (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $O = I_{0} \times Q \times R^{3 N}$ is open and $(O, L)$ is a time-dependent Lagrangian chart whenever $Q \neq \emptyset$ and $a^{'} < b^{'}$ with $[a^{'}, b^{'}] \subseteq I_{0}$ , over $[a^{'}, b^{'}] \times Q$ ; $L = K - U$ with $L_{x} = - \nabla_{x} U$ , $L_{v} = M v$ ; $F_{i} = - \nabla_{x_{i}} U + d_{i}$ ; and $U$ is $C^{2}$ so that the map $(t, x) \mapsto \sum_{i} a \cdot \nabla_{x_{i}} U (t, x)$ is continuous (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F2]

The model's symmetry vocabulary (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): a $C^{2}$ triple $(τ, ξ, B)$ is a symmetry datum exactly when $L_{t} τ + L_{x} \cdot ξ + L_{v} \cdot (D_{t} ξ - v D_{t} τ) + L D_{t} τ = D_{t} B$ on $O$ , with charge $(M v) \cdot ξ - E τ - B$ and $E = K + U$ ; for $τ \equiv 0$ the identity is $L_{x} \cdot ξ + L_{v} \cdot D_{t} ξ = D_{t} B$ and the charge is $(M v) \cdot ξ - B$ .

[F3]

[F4]

The translation statement (Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness clauses 1 and 3): the differential condition is equivalent to potential invariance along admissible segments; invariance across excluded segments also requires agreement of the interval-component constants. Under the differential condition with $d \equiv 0$ , $a \cdot P$ is constant along every motion.

[F5]

Momentum, centre of mass and motions (Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance): $P = \sum_{i} m_{i} x_{i}^{'}$ , $M = \sum_{i} m_{i}$ is the scalar total mass and $M X = \sum_{i} m_{i} x_{i}$ , so $M X^{'} = P$ ; along a motion $M x^{''} = F$ and $P^{'} = F_{t o t} : = \sum_{i} F_{i}$ , and through every initial datum there is exactly one maximal motion with open maximal interval containing the initial time.

[F6]

Calculus (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ): sums, scalar multiples and pairings differentiate by the usual rules; a $C^{1}$ function on an interval is constant exactly when its derivative vanishes. For $C^{2}$ $U$ , differentiating the vanishing directional derivative gives $\nabla_{x} (a \cdot \nabla_{x} U) = 0$ , so the spatial gradient is constant along every translation segment contained in $Q$ . Global finite invariance likewise gives equality of the endpoint gradients by differentiating its identity on an open neighbourhood of the two admissible endpoints.

[F7]

Parameter units (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions): $[s] = ℓ τ^{- 1}$ and $[a] = 1$ , so $[K_{a}] = μ ℓ$ and $[K_{a}^{'}] = μ ℓ τ^{- 1}$ , as also follows from $K_{a}^{'} = t a \cdot F_{t o t}$ .

Proof

technique · direct; evaluate the model identity at the boost triple and reduce its validity to the translation condition by continuity in time, transfer conservation through the model's general Noether theorem, verify the boosted curves, and compute the charge rate from the equation of motion

1.1F1F2F4F6algebra

Clause 1, evaluation of the identity: for the triple $(0, ξ^{a}, B^{a})$ one has $D_{t} ξ^{a} = (a, \dots, a)$ , $L_{x} \cdot ξ^{a} = - t \sum_{i} a \cdot \nabla_{x_{i}} U (t, x)$ and $L_{v} \cdot D_{t} ξ^{a} = \sum_{i} m_{i} a \cdot v_{i} = D_{t} B^{a}$ , since $D_{t} B^{a} = 0 + \sum_{i} m_{i} a \cdot v_{i}$ ; hence the two sides of the strict identity are $t g (t, x)$ apart, with $g (t, x) : = \sum_{i} a \cdot \nabla_{x_{i}} U (t, x)$ , and the identity holds on $O$ if and only if $t g (t, x) = 0$ for all $(t, x) \in I_{0} \times Q$ . Since $g$ is continuous and $I_{0}$ contains points $t \neq 0$ , this holds if and only if $g (t, x) = 0$ for every $t \neq 0$ in $I_{0}$ and every $x \in Q$ ; and if $0 \in I_{0}$ then continuity of $g$ at $(0, x)$ forces $g (0, x) = 0$ for every $x \in Q$ . Hence the identity is equivalent to $g \equiv 0$ on $I_{0} \times Q$ , which is the translation-invariance condition of [F4].

1.2F1F5F6algebra

Clause 5, the rate: along a motion, $t \mapsto K_{a} (t) = t (a \cdot P (t)) - M a \cdot X (t)$ is $C^{1}$ by [F5] and [F6], and differentiating gives $K_{a}^{'} = a \cdot P + t a \cdot P^{'} - M a \cdot X^{'} = a \cdot P + t a \cdot F_{t o t} - a \cdot P = t a \cdot F_{t o t},$ because $M X^{'} = P$ and $P^{'} = F_{t o t} = \sum_{i} F_{i} (t, x, x^{'})$ along the motion. In the case $d \equiv 0$ one has $F_{i} (t, x, x^{'}) = - \nabla_{x_{i}} U (t, x)$ , so $K_{a}^{'} = - t g (t, x (t))$ and, $K_{a}$ being $C^{1}$ on the interval $J$ , it is constant exactly when $t g (t, x (t))$ vanishes identically on $J$ ; if $g$ is not identically zero on $I_{0} \times Q$ , continuity of $g$ supplies a point $(t_{0}, x_{0})$ with $t_{0} \neq 0$ and $g (t_{0}, x_{0}) \neq 0$ (a point with $g \neq 0$ at time $0$ has a neighbourhood on which $g \neq 0$ , and $I_{0}$ contains times $\neq 0$ ), and the maximal motion through $(t_{0}, x_{0}, 0)$ has $K_{a}^{'} (t_{0}) = - t_{0} g (t_{0}, x_{0}) \neq 0$ , so $K_{a}$ is not constant on its interval.

1.3F1F4F5F6algebra

Clause 4, boosted motions: with $d \equiv 0$ , the curve $\tilde{x} = x + s t a$ is $C^{2}$ , takes values in $Q$ by hypothesis and satisfies ${\tilde{x}}^{''} = x^{''}$ . Its motion equation $M {\tilde{x}}^{''} = - \nabla_{x} U (t, \tilde{x})$ is therefore equivalent to $\nabla_{x} U (t, \tilde{x}) = \nabla_{x} U (t, x)$ , since $x$ is a motion. If every joining segment lies in $Q$ , the derivative along that segment of $\nabla_{x} U$ is $\nabla_{x} (a \cdot \nabla_{x} U) = 0$ by the differential hypothesis and [F6], giving the equality. Alternatively, under global finite invariance, for fixed $t$ and shift $s t a$ both endpoints have an open neighbourhood on which $U (t, y + s t a) = U (t, y)$ ; differentiation with respect to $y$ gives the same gradient equality. Thus both stated sufficient routes produce a boosted motion, and the exact criterion holds without either extra route.

2.1F2F3step 1.1

Clause 2, conservation: by step 1.1 and the hypothesis, the datum $(0, ξ^{a}, B^{a})$ satisfies the model identity on $O$ , so with $d \equiv 0$ clause 1 of [F3] gives that its charge is constant along every motion; by [F2] that charge is $(M x^{'} (t)) \cdot t a - (K (x^{'} (t)) + U (t, x (t))) \cdot 0 - B^{a} (t, x (t)) = t (a \cdot P (t)) - M a \cdot X (t) = K_{a} (t)$ .

2.2F6step 1.2

Clause 5, the non-constancy statement: by step 1.2, in the case $d \equiv 0$ a point $(t_{0}, x_{0})$ with $t_{0} \neq 0$ and $g (t_{0}, x_{0}) \neq 0$ yields a maximal motion with $K_{a}^{'} (t_{0}) = - t_{0} g (t_{0}, x_{0}) \neq 0$ , and by [F6] a function constant on the interval $J_{\max}$ would have vanishing derivative at $t_{0}$ .

3.1F4step 2.1algebra

Clause 3: by step 2.1 the function $t \mapsto K_{a} (t) = t (a \cdot P (t)) - M a \cdot X (t)$ is constant on $J$ , and by [F4] the slope $a \cdot P$ is constant on $J$ ; hence $M a \cdot X (t) = t (a \cdot P) - K_{a}$ is affine in $t$ with slope $a \cdot P$ on $J$ .

4.1step 1.1step 1.2step 1.3step 2.1step 2.2step 3.1F7∎

Clause 1 is step 1.1, clause 2 is step 2.1, clause 3 is step 3.1, clause 4 is step 1.3, and the two statements of clause 5 are step 1.2 and step 2.2; the charge is conserved exactly under the potential invariance that conserves the momentum component, and no boost invariance of any particular force law and no empirical claim is asserted. By [F7], $[K_{a}] = μ ℓ$ and $[K_{a}'] = μ ℓ τ^{- 1}$ , the charge and rate dimensions for a velocity-valued boost parameter.

Remarks

Kinematics versus model symmetry. Every model of the framework is described in every inertial frame, and the Galilean transformation of the description is available whatever the force law (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (vi)); the boost charge, by contrast, is conserved in the global potential branch when the differential translation condition of Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness holds for its generator. For the one-label confining oscillator $U = \frac{k}{2} ∣ x ∣^{2}$ , $k > 0$ with units $μ τ^{- 2}$ , the global translation condition fails for every nonzero direction $a$ (evaluate it at a nonzero position parallel to $a$ ). At positions with $a \cdot x \neq 0$ the directional derivative is nonzero, and in the global potential branch the centre of mass oscillates rather than moves uniformly. This is the precise boundary of the boost statement: it is a property of the model's Lagrangian, not of Galilean kinematics.
Relation to the momentum clause. Clauses 1 and 2 show that for the global potential branch the boost charge adds no independent conservation hypothesis beyond the differential translation condition already required by Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness: the two charges $a \cdot P$ and $K_{a}$ stand or fall together, and clause 3 says that the content of the boost charge is the uniform motion of the component $a$ of the centre of mass. With residual forces the rate formula of clause 5 replaces conservation, and the charge fails to be conserved along every motion on which the component $a$ of the total force does not vanish identically.
Empirical status. All statements are deductions inside the model under the declared hypotheses; the phrase "boost" names the transformation $x_{i} \mapsto x_{i} + s t a$ of the description, not any measurement, and no claim is made that a physical force law is boost invariant.

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The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions

Physical scope

The energy bookkeeping of a time-dependent coordinate description of the force model: for a C^3 coordinate map X with everywhere invertible configuration Jacobian and the substituted Lagrangian of the covariance theorem, the chart energy function equals the inertial mechanical energy plus or minus the pairing of the chart's explicit time dependence with the mass-operator velocity, so the two agree exactly when that pairing vanishes; the chart energy function is conserved along chart Euler-Lagrange solutions whenever the substituted Lagrangian has no explicit time dependence along the trajectory; and in the global potential branch the chart description of every motion solves that chart equation. The statement concerns the model's coordinate description, is conditional on the adopted force branch and the declared presentation, and asserts no measurement and no identification of the chart energy with the inertial mechanical energy.

Statement

$\tilde{O} : = {(t, q, v) \in Ω \times R^{3 N} : (t, X (t, q), X_{t} (t, q) + X_{q} (t, q) v) \in O}, \tilde{L} (t, q, v) : = L (t, X (t, q), X_{t} (t, q) + X_{q} (t, q) v)$

be the substituted data of Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation. Let $J \subseteq R$ be an interval with nonempty interior, with derivatives at any included endpoints understood one-sided, and let $q : J \to R^{3 N}$ be a $C^{2}$ curve with $(t, q (t), q^{'} (t)) \in \tilde{O}$ for every $t \in J$ , and put $x (t) : = X (t, q (t))$ . Define the chart energy function and the inertial mechanical energy along the curve by

$e (t) : = q^{'} (t) \cdot {\tilde{L}}_{v} (t, q (t), q^{'} (t)) - \tilde{L} (t, q (t), q^{'} (t)), E_{i n} (t) : = K (x^{'} (t)) + U (t, x (t)) .$

Then:

(difference identity). For every $t \in J$ ,

$E_{i n} (t) = e (t) + ⟨ X_{t} (t, q (t)), M x^{'} (t) ⟩;$

equivalently $e (t) = \frac{1}{2} ⟨ X_{q} v, M X_{q} v ⟩ - \frac{1}{2} ⟨ X_{t}, M X_{t} ⟩ + U (t, X)$ at $(t, q (t), q^{'} (t))$ . 2. (coincidence criterion). $e (t) = E_{i n} (t)$ if and only if $⟨ X_{t} (t, q (t)), M x^{'} (t) ⟩ = 0$ ; in particular $e = E_{i n}$ along the curve whenever $X_{t} (t, q (t)) = 0$ . 3. (conservation of the chart energy). If ${\tilde{L}}_{t} (t, q (t), q^{'} (t)) = 0$ for every $t \in J$ and $q$ solves the chart Euler-Lagrange equation $E_{\tilde{L}} (q) = 0$ on $J$ , then $e$ is constant on $J$ ; this includes in particular the case ${\tilde{L}}_{t} \equiv 0$ on $\tilde{O}$ . 4. (model motions in the global potential branch). If $d \equiv 0$ and $x : J \to Q$ is a motion of the model with $q$ its chart description, then $E_{\tilde{L}} (q) = 0$ on $J$ .

The difference identity and the coincidence criterion hold for every presentation; clauses 3 and 4 are the conservation statement of the moving description and its link to the model's motions. No identification of the chart energy function with the inertial mechanical energy is asserted, and no empirical content is claimed.

Facts & Assumptions

Given: The model, the presentation $(U, d)$ , the coordinate map $X$ , the substituted data $\tilde{O}, \tilde{L}$ , and the curve $q$ with $x = X \circ q$ .

[F1]

The substituted chart (Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations): $\tilde{O}$ is open, $\tilde{L}$ is $C^{2}$ on it, along the curve $E_{\tilde{L}} (q) = X_{q} (t, q (t))^{T} E_{L} (x) (t)$ with $x = X \circ q$ , and if $d \equiv 0$ and $x$ is a motion then $E_{L} (x) = 0$ and hence $E_{\tilde{L}} (q) = 0$ ; this pointwise identity is applied on nondegenerate closed subintervals of $J^{\circ}$ and extended to included endpoints by continuity; also $x^{'} (t) = X_{t} (t, q (t)) + X_{q} (t, q (t)) q^{'} (t)$ .

[F2]

The mechanical Lagrangian (The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator): $L (t, x, v) = K (v) - U (t, x)$ with $K (v) = \frac{1}{2} ⟨ v, M v ⟩$ , $M$ symmetric positive definite, $U$ of class $C^{2}$ , $L_{v} = M v$ , $L_{x} = - \nabla_{x} U$ ; the force map is $F_{i} = - \nabla_{x_{i}} U + d_{i}$ , and for $d \equiv 0$ the motions satisfy $M x^{''} = - \nabla_{x} U$ .

[F3]

Mechanical chart Lagrangians (A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian): for $C^{2}$ fields $G, A, V$ on an open $J_{0} \times U_{0}$ with $G (t, q)$ symmetric positive definite, the Lagrangian $L = \frac{1}{2} v^{T} G v + A \cdot v - V$ is $C^{2}$ with $L_{v} = G v + A$ , $L_{v v} = G$ , and is hyperregular with inverse Legendre branch $v = G^{- 1} (p - A)$ in the sense of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity.

[F4]

The hyperregular energy identity (The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity): for a $C^{2}$ curve on an open interval $I$ satisfying $E_{L} (q) = 0$ on a hyperregular chart, the function $e = L_{v} \cdot v - L$ satisfies $e^{'} (t) = - L_{t}$ on $I$ ; if $L_{t}$ vanishes along the trajectory, then $e$ is constant on $I$ .

[F5]

Calculus and linear algebra (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The transpose $A^{T}$ of a matrix, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, Linear map between vector spaces over the same field, Invertible Euclidean linear maps, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): the Euclidean inner product is symmetric and bilinear; for a linear map $A$ one has $⟨ A u, w ⟩ = ⟨ u, A^{T} w ⟩$ ; the composite $x = X \circ q$ is $C^{2}$ with the displayed derivative; and $⟨ X_{q} v, M X_{q} v ⟩ > 0$ for $v \neq 0$ because $X_{q}$ is injective and $M$ is positive definite.

Proof

technique · direct; expand the substituted Lagrangian and its velocity derivative, compare with the inertial mechanical energy, verify the mechanical-chart hypotheses for the substituted Lagrangian, and apply the hyperregular energy identity

1.1F1F2F5algebra

Clause 1, expansion: since $x^{'} (t) = X_{t} + X_{q} q^{'}$ by [F1], the definition of $\tilde{L}$ gives $\tilde{L} = \frac{1}{2} ⟨ X_{t} + X_{q} v, M (X_{t} + X_{q} v) ⟩ - U (t, X) = \frac{1}{2} v^{T} G v + A \cdot v + \frac{1}{2} ⟨ X_{t}, M X_{t} ⟩ - U (t, X)$ with $G = X_{q}^{T} M X_{q}$ and $A = X_{q}^{T} M X_{t}$ , and ${\tilde{L}}_{v} = G v + A = X_{q}^{T} M (X_{t} + X_{q} v)$ ; hence $e = ⟨ v, {\tilde{L}}_{v} ⟩ - \tilde{L} = \frac{1}{2} ⟨ X_{q} v, M X_{q} v ⟩ - \frac{1}{2} ⟨ X_{t}, M X_{t} ⟩ + U (t, X)$ at $(t, q (t), q^{'} (t))$ , while $E_{i n} = \frac{1}{2} ⟨ X_{t} + X_{q} v, M (X_{t} + X_{q} v) ⟩ + U (t, X);$ subtracting and expanding the square gives $E_{i n} - e = ⟨ X_{t}, M (X_{t} + X_{q} v) ⟩ = ⟨ X_{t}, M x^{'} ⟩$ , which is the displayed difference identity.

1.2F1F2given

Clause 4: by [F1], along the curve $q$ one has $E_{\tilde{L}} (q) = X_{q}^{T} E_{L} (x)$ ; if $d \equiv 0$ and $x$ is a motion, then $E_{L} (x) = 0$ by [F1] and [F2], so $E_{\tilde{L}} (q) = 0$ on $J$ .

2.1F2F3F5step 1.1

Clause 3, local hyperregularity: put $D : = Ω \cap (I_{0} \times R^{3 N})$ , so $\tilde{O} = D \times R^{3 N}$ . By step 1.1, $\tilde{L} = \frac{1}{2} v^{T} G v + A \cdot v - V$ with $V : = U (t, X) - \frac{1}{2} ⟨ X_{t}, M X_{t} ⟩$ . The fields $G, A, V$ are $C^{2}$ on $D$ because $X$ is $C^{3}$ and $U$ is $C^{2}$ , and $G$ is symmetric positive definite by [F5]. Around each $(t, q) \in D$ choose an open product $J_{0} \times U_{0} \subseteq D$ , with $J_{0}$ an interval and $U_{0}$ a ball. On this product [F3] applies, giving hyperregularity and the $C^{2}$ inverse $v = G^{- 1} (p - A)$ . These inverses agree on overlaps because the displayed formula is unique in each fibre; thus it defines a $C^{2}$ inverse on $D \times R^{3 N}$ , and each product restriction is a hyperregular chart in the precise sense of [F3].

2.2step 1.1

Clause 2: by the difference identity of step 1.1, $e (t) = E_{i n} (t)$ holds exactly when $⟨ X_{t} (t, q (t)), M x^{'} (t) ⟩ = 0$ ; if $X_{t} (t, q (t)) = 0$ the pairing vanishes identically along the curve.

3.1F4F5step 2.1

Clause 3, conservation: $J^{\circ}$ is a nonempty open interval. For each $t_{0} \in J^{\circ}$ , choose a product neighborhood $J_{0} \times U_{0} \subseteq D$ of $(t_{0}, q (t_{0}))$ as in step 2.1. Continuity of $q$ gives an open interval $I ∋ t_{0}$ inside $J^{\circ} \cap J_{0}$ with $q (I) \subseteq U_{0}$ . On that restriction [F4] applies and gives $e^{'} (t_{0}) = - {\tilde{L}}_{t} (t_{0}, q (t_{0}), q^{'} (t_{0})) = 0$ . Hence $e^{'} = 0$ throughout $J^{\circ}$ and $e$ is constant there by [F5]. The defining formula and $C^{2}$ regularity make $e$ continuous on $J$ , so the same constant extends to every included endpoint.

4.1step 1.1step 2.1step 2.2step 3.1∎

Clause 1 is step 1.1, clause 2 is step 2.2, clause 3 is step 3.1, and clause 4 is step 1.2; the chart energy function of a moving description is conserved along the chart solutions whenever the substituted Lagrangian has no explicit time dependence along the trajectory, and it coincides with the inertial mechanical energy exactly when the explicit time dependence pairs to zero with the mass-operator velocity.

Remarks

The moving-chart boundary. The identity of clause 1 is the exact bookkeeping behind the phrase "in a moving chart the Hamiltonian need not equal the mechanical energy": for a chart with $X_{t} \equiv̸ 0$ the two functions differ by $⟨ X_{t}, M x^{'} ⟩$ , which is not constant in general, and clause 3 shows that the chart energy function can nevertheless be a conserved quantity of the chart equations. For a component of the model's motion the chart curve solves the chart equation in the global potential branch by clause 4, so a conserved chart energy that differs from the inertial mechanical energy is a genuine feature of the model's description and not a computational artefact. The concrete rotating-chart computation of the companion page evaluates both functions on explicit data.
Relation to the Hamiltonian. On a hyperregular chart the chart energy function equals the branch Hamiltonian $H = p \cdot v - \tilde{L}$ along the solutions (The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity clause 3), so clause 3 is the statement that $H$ is a first integral of the chart equations when no explicit time dependence is present; this is the chart-level reading of the energy case of Noether identity for point transformations with time reparametrisation: the conserved charge $p \cdot ξ - E τ - B$ , applied to the substituted Lagrangian rather than to the inertial one.
Empirical status. All clauses concern the model's coordinate description and are conditional deductions from the adopted force branch, the declared presentation and the smooth invertibility of $X$ ; no measurement is reported and no chart quantity is identified with a measured energy.

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Scope of the symmetry-conservation statements: force laws that break kinematic symmetries, residual forces, moving coordinates and the empirical boundary

Physical scope

Scope audit of the symmetry and conservation statements of the pair: the symmetry criteria are conditions on the declared model Lagrangian and potential, not on the kinematic framework; in the global potential branch (d=0), force laws with external or confining potentials can break the corresponding charges, with the failure rates computed by the sharpness clauses; for arbitrary presentations, residual-force terms can cancel or add to those rates, and the exact rates are supplied by the Noether statements; in a moving coordinate description the conserved chart energy function need not be the inertial mechanical energy; explicit time dependence breaks energy conservation in the global potential branch when it is nonzero at an adopted state; and no converse from an observed conservation to a symmetry of a model, and no empirical claim, is asserted. The remark adds no new physical claim.

Remarks

Symmetries are properties of the model, not of the kinematics. The symmetry criteria of this page are identities imposed on the model's mechanical Lagrangian $L = K - U$ of the declared potential presentation; residual forces enter the charge rates separately (Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions); the affine structure of spacetime, the availability of every inertial frame and the Galilean transformation law do not by themselves make a potential invariant. For the uniform-force potential $U = - \sum_{i} m_{i} g \cdot x_{i}$ with $d \equiv 0$ , $\sum_{i} \nabla_{x_{i}} U = - M_{t o t} g$ : the strict translation datum $τ = 0$ , $ξ_{i} = a$ , $B = 0$ satisfies the identity exactly in nonzero dimensionless directions $a ⊥ g$ , and a direction with $a \cdot g \neq 0$ gives a momentum and boost charge that fail along motions. The sharpness clauses locate that failure along maximal motions through states where their required directional derivative is nonzero.

A mass-weighted confining potential $U = \frac{k}{2} \sum_{i} m_{i} ∣ x_{i} - x_{i}^{0} ∣^{2}$ , $k > 0$ , has $\sum_{i} \nabla_{x_{i}} U = k M (X - X_{0})$ , where $X_{0} = M^{- 1} \sum_{i} m_{i} x_{i}^{0}$ . In the global potential model, for any direction with $a \cdot (X - X_{0}) \neq 0$ the translation condition fails and the momentum charge is not conserved along the maximal motion through that state. If that state's time is nonzero, the boost sharpness clause also applies; at time zero, continuity gives nearby nonzero times with the same nonvanishing directional derivative. Here $k$ has units $τ^{- 2}$ , so $U$ has energy units and the centre-of-mass equation is $X^{''} = - k (X - X_{0})$ .
Rotation is not translation. The rotation condition of Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin is compatible with a confining central potential: for $U$ depending only on the labels through rotation-invariant combinations the angular-momentum charge for a dimensionless rotation generator may be conserved in the global potential branch even when the translation charges are not. The conservation statements are separate conditional claims, one per symmetry datum; a model can possess one charge and fail another.
Residual forces destroy charges at a computable rate. With $d \equiv̸ 0$ the charge of a strict symmetry is conserved exactly when the residual force pairs to zero with the generator along the motion (The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate clause 2), and for a general datum with time reparametrisation the rate is the residual force paired with $ξ - v τ$ (The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p \cdot ξ - E τ - B$ , its residual-force rate and its potential gauge clause 3). A friction force along a translation generator, for instance, changes the corresponding momentum charge whenever its total pairing with that generator is nonzero along the motion. The balance theorems of the force page, in particular Mechanical energy balance for a potential presentation: $E^{'} = \partial_{t} U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form, remain valid in those situations and carry the residual contribution explicitly; the Noether charges and the balance identities are different deductions, and neither replaces the other.
Moving coordinates and the mechanical energy. In a time-dependent coordinate description the chart energy function of The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions differs from the inertial mechanical energy by the pairing of the chart's explicit time dependence with the mass-operator velocity, and the chart energy can be conserved while that difference is nonzero; the phrase "the Hamiltonian is conserved" is therefore not by itself a statement about the inertial mechanical energy. For the datum $(τ, ξ, B) = (1, 0, 0)$ , with parameter measured in seconds, the time-translation identity of Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential holds exactly when $\partial_{t} U \equiv 0$ on $I_{0} \times Q$ . If $\partial_{t} U \equiv̸ 0$ , this datum fails the identity; in the global potential branch the sharpness clause exhibits a maximal motion through a state where $\partial_{t} U \neq 0$ whose mechanical energy has that nonzero derivative there.
No converse and no empirical claim. The results of this page are conditional deductions inside the force model: symmetry hypotheses on the declared presentation imply conservation of the stated charges along the motions. No converse is asserted — conservation of a quantity along the computed motions does not establish that the model has the corresponding symmetry datum — and no observational content is introduced. If a conservation statement is used in an empirical analysis, the model assumptions, the representative of the potential and any measured inputs enter that analysis as additional premises with their own qualifications; an observation consistent with a conservation law is evidence about the combined model and apparatus, not a proof of either.
Not asserted. The page does not treat discrete symmetries, Noether's second theorem, field-theoretic charges, gauge families whose generators fail to be $C^{2}$ , global existence of the transformation families, or the question whether any physical interaction is invariant under the families considered; those are separate mathematical and empirical developments.

5 · Examples, experiments and counterexamples

None yet.

Symmetry, Noether and conditional conservation

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions

Definition

Remarks

The point Noether theorem of the model: conservation of the charge of a strict symmetry, and the residual-force rate

Statement

Facts & Assumptions

Proof

Remarks

The general Noether theorem of the model: point transformations with time reparametrisation, the charge p⋅ξ−Eτ−Bp⋅ξ−Eτ−B, its residual-force rate and its potential gauge

Statement

Facts & Assumptions

Proof

Remarks

Time translation and the mechanical energy of the model: conservation in the autonomous global potential branch and the sharp failure for a time-dependent potential

Statement

Facts & Assumptions

Proof

Remarks

Translation symmetry of the model's Lagrangian and conservation of total momentum: exact hypothesis, conservation and its sharpness

Statement

Facts & Assumptions

Proof

Remarks

Rotation symmetry of the model's Lagrangian and conservation of angular momentum about the origin

Statement

Facts & Assumptions

Proof

Remarks

The Galilean boost charge of the model: exact condition, uniform centre-of-mass motion, the induced motions and the boundary of the conservation law

Statement

Facts & Assumptions

Proof

Remarks

The energy function of a moving chart: exact difference from the inertial mechanical energy and conservation along the chart solutions

Statement

Facts & Assumptions

Proof

Remarks

Scope of the symmetry-conservation statements: force laws that break kinematic symmetries, residual forces, moving coordinates and the empirical boundary

Remarks

5 · Examples, experiments and counterexamples

Sources

The general Noether theorem of the model: point transformations with time reparametrisation, the charge $p \cdot ξ - E τ - B$ , its residual-force rate and its potential gauge