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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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

Depends on

Dependency tree · 2 levels

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