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The velocity Hessian of the mechanical Lagrangian and the local invertibility of its Legendre map in every chart of the model

Physical scope

The Legendre map of a declared potential presentation of the force branch: the velocity Hessian of the mechanical Lagrangian in an inertial Cartesian chart and in a declared generalized-coordinate chart, its positivity, the resulting velocity-regularity at every state, and the existence of a local C^1 inverse branch at every state of those charts. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it asserts no global inverse branch on a momentum fibre, no equation of motion, no conservation law and no measured value.

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 the data and conventions of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta: masses $m_{1}, \dots, m_{N} > 0$ , open configuration domain $Q \subseteq R^{3 N}$ , time window $I_{0}$ , mass operator $M$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^{2}$ potential $U : I_{0} \times Q \to R$ and residual forces $d$ of the standing regularity (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). Let $X : Ω \to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ and kinetic metric $G (q) = D X (q)^{T} M D X (q)$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). Then:

(velocity Hessian) For every $(t, x, v) \in I_{0} \times Q \times R^{3 N}$ one has $L_{v} (t, x, v) = M v$ and $L_{v v} (t, x, v) = M$ , and for every $(t, q, v) \in I_{0} \times Ω \times R^{r}$ one has ${\tilde{L}}_{v} (t, q, v) = G (q) v$ and ${\tilde{L}}_{v v} (t, q, v) = G (q)$ . Both matrices are symmetric and positive definite; consequently the Cartesian presentation and the presentation in the coordinates $X$ are velocity-regular at every state in the sense of Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts.
(local inverse branch) At every state $z$ of either chart there is a local inverse branch: open sets $W ∋ z$ and $W^{'} ∋ F (z)$ with $F ∣_{W} : W \to W^{'}$ a $C^{1}$ diffeomorphism, where $F$ is the Legendre map of the corresponding fibre-full chart (Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts clause (c), The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity).

The statement is conditional on the adopted force branch and on the declared presentation; it asserts no global inverse Legendre branch on a momentum fibre and no empirical content.

Facts & Assumptions

Given: The model data of the statement, the mechanical Lagrangian $L = K - U$ , the generalized-coordinate map $X$ with its substituted Lagrangian, and a state of one of the two charts.

[F1]

Model and Lagrangian data (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): $L = K - U$ is of class $C^{2}$ on the open set $I_{0} \times Q \times R^{3 N}$ , the velocity gradient and Hessian are formed with the Euclidean partial derivatives of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $K^{'} (v) = M v$ , the mass operator $M$ is a symmetric positive definite linear map of $R^{3 N}$ , and the presentation $(U, d)$ has $U$ of class $C^{2}$ with the standing regularity of the force branch; dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework.

[F2]

Mechanical chart Lagrangian (A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): for $C^{2}$ data $G, A, V$ with $G (t, q)$ symmetric positive definite, the Lagrangian $L (t, q, v) = \frac{1}{2} v^{T} G (t, q) v + A (t, q) \cdot v - V (t, q)$ has $L_{v} (t, q, v) = G (t, q) v + A (t, q)$ and $L_{v v} (t, q, v) = G (t, q)$ , and is regular on its chart; the products and transposes are those of the cited matrix items and the dot product is the Euclidean pairing.

[F3]

Generalized-coordinate dictionary (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces): the substituted Lagrangian is $\tilde{L} (t, q, v) = L (t, X (q), D X (q) v) = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q)$ with $\tilde{U} (t, q) = U (t, X (q))$ , the map $X$ is of class $C^{3}$ with injective derivative, and $G (q)$ is symmetric and positive definite.

[F4]

Legendre derivative and inverse function theorem (Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse, The Euclidean inverse function theorem): for a fibre-full $C^{2}$ chart the Legendre map is $C^{1}$ , its derivative acts by $(τ, δ q, δ v) \mapsto (τ, δ q, L_{v t} τ + L_{v q} δ q + L_{v v} δ v)$ , it is invertible exactly when the velocity Hessian is nonsingular, and a nonsingular velocity Hessian yields a local $C^{1}$ diffeomorphism onto the image; the cited inverse function theorem supplies the local $C^{1}$ inverse for a $C^{1}$ map with invertible derivative.

[F5]

Fibre-full charts and closure of differentiability (The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, $C^{k}$ Euclidean maps and diffeomorphisms): a fibre-full time-dependent Lagrangian chart is a $C^{2}$ map on an open set of the form $J \times U \times R^{n}$ ; finite sums and products of $C^{k}$ Euclidean maps are $C^{k}$ and composites of composable $C^{k}$ maps are $C^{k}$ .

Proof

technique · direct; identify the model's two chart Lagrangians with mechanical chart Lagrangians, read off the velocity Hessians, and apply the local inverse-function criterion

1.1F1F5algebra

The set $I_{0} \times Q \times R^{3 N}$ is open, and $L$ is a $C^{2}$ map on it by [F1]; hence $(I_{0} \times Q \times R^{3 N}, L)$ is a fibre-full time-dependent Lagrangian chart in the sense of [F5]. By [F1] one has $L (t, x, v) = \frac{1}{2} v \cdot M v - U (t, x)$ with $M$ symmetric positive definite and $U$ of class $C^{2}$ , so $L$ is the mechanical chart Lagrangian of [F2] with the data $G : = M$ , $A : = 0$ , $V : = U$ .

1.2F2F3F5algebra

By [F3] the substituted Lagrangian is $\tilde{L} (t, q, v) = L (t, X (q), D X (q) v) = \frac{1}{2} v^{T} G (q) v - \tilde{U} (t, q)$ with $\tilde{U} (t, q) = U (t, X (q))$ ; since $X$ is of class $C^{3}$ its derivative $D X$ is of class $C^{2}$ , so $G (q) = D X (q)^{T} M D X (q)$ and $\tilde{U}$ are of class $C^{2}$ by the closure properties of [F5], and $G (q)$ is symmetric positive definite by [F3]. Thus $\tilde{L}$ is the mechanical chart Lagrangian of [F2] with the data $G : = G (\cdot)$ , $A : = 0$ , $V : = \tilde{U}$ , and [F2] gives ${\tilde{L}}_{v} (t, q, v) = G (q) v$ and ${\tilde{L}}_{v v} (t, q, v) = G (q)$ .

1.3F3algebra

By [F3] the determinant $\det G (q)$ is nonzero at every $q$ , because $G (q)$ is positive definite and positive definiteness implies invertibility; hence the presentation in the coordinates $X$ is velocity-regular at every state.

2.1F1F2step 1.1algebra

Applying [F2] to step 1.1 gives $L_{v} (t, x, v) = M v + 0 = M v$ and $L_{v v} (t, x, v) = M$ for every $(t, x, v)$ ; since $M$ is symmetric positive definite by [F1], the Cartesian presentation is velocity-regular at every state in the sense of Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts.

2.2F4step 1.2step 1.3algebra

Generalized-coordinate local branch: the same argument applies to the chart of step 1.2 with the nonsingular velocity Hessian $G (q)$ of step 1.3, and gives a local inverse branch at every state of the chart of $X$ .

3.1F4step 2.1algebra

Cartesian local branch: by [F4] the Legendre map $F$ of the chart of step 1.1 is $C^{1}$ , its derivative at a state is invertible exactly when the velocity Hessian there is nonsingular, and by step 2.1 the velocity Hessian is the invertible matrix $M$ ; hence $F$ has invertible derivative at every state, and [F4] (the inverse function theorem applied to the $C^{1}$ map $F$ at the state) yields open sets $W ∋ (t, x, v)$ and $W^{'} ∋ F (t, x, v)$ with $F ∣_{W} : W \to W^{'}$ a $C^{1}$ diffeomorphism.

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

Clause 1 follows from steps 2.1, 1.2 and 1.3 together with the definition of velocity-regularity; clause 2 follows from steps 3.1 and 2.2 together with the notion of local inverse branch; no statement about a global inverse branch has been used.

Remarks

What is used and what is not. The proof uses only the positivity of the mass operator and of the kinetic metric, the identification of the model's Lagrangian with a mechanical chart Lagrangian, and the local inverse-function criterion of Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse; it does not use hyperregularity, and it does not claim that the local branches globalise. The global statement for the mechanical branch is proved later on this page, and the companion example $L (v) = e^{v}$ shows that regularity at every state can fail to give a global fibre inverse.
Source reading. The worked case $p = m \dot{r}$ , $H = p^{2} / (2 m) + V (r)$ of Tong's Section 4.1.3 is the Cartesian instance of clauses 1 and 2; the general quadratic form with positive definite kinetic matrix appears in Stewart's Section 1.2, and SICM's Section 3.1.1 states explicitly that the Legendre transform is available in closed form for velocity-quadratic Lagrangians because the velocity derivative is then invertible. The source audit's caveat that convexity is not the relevant hypothesis is respected: the hypothesis used here is the nonsingularity recorded in clause 1.

The velocity Hessian of the mechanical Lagrangian and the local invertibility of its Legendre map in every chart of the model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources