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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

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

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

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.

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

Depends on

Dependency tree · 2 levels

Sources