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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

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. Let $Ω \subseteq R \times R^{3 N}$ be open and nonempty and let $X : Ω \to Q$ be of class $C^{3}$ with $X_{q} (t, q)$ invertible for every $(t, q) \in Ω$ ; let

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

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.

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

Depends on

Dependency tree · 2 levels

Sources