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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources