Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts

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 in the fixed inertial frame with adapted coordinates $(t,x)$: masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator), $C^2$ potential $U$ 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). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, and the Euclidean gradients and second partial derivatives in the velocity slot are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$.

(a) Velocity Hessian. When $Q\mathrm{\ne }\mathrm{\varnothing }$, let $(O,L)$ be the fibre-full time-dependent Lagrangian chart of clause (c) of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions and let $\tilde{L}$ be the substituted Lagrangian of a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). The velocity Hessian of the presentation in Cartesian data is the matrix $L_{vv}(t,x,v)\in M_{3N}(\mathbb{R})$ of second partial derivatives of $L$ in its third slot at $(t,x,v)$; in the coordinates $X$ it is the matrix ${\tilde{L}}_{vv}(t,q,v)$ of second partial derivatives of $\tilde{L}$ in its third slot. Invertibility of the Hessian is invertibility of a linear map in the sense of Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$, read by A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit through the determinant, and matrices act on vectors by Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes and Linear map between vector spaces over the same field; transposes are those of The transpose $A^{\mathsf{T}}$ of a matrix.

(b) Regular states and regular presentations. The presentation is velocity-regular at the state $(t,x,v)$ of the Cartesian chart when $\det L_{vv}(t,x,v)\mathrm{\ne }0$, and velocity-regular at the state $(t,q,v)$ of the chart of $X$ when $\det {\tilde{L}}_{vv}(t,q,v)\mathrm{\ne }0$. It is regular on a chart when it is velocity-regular at every state of that chart, and regular when this holds on the chart under discussion. This is exactly the regularity of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity instantiated, for $Q\mathrm{\ne }\mathrm{\varnothing }$, at the corresponding fibre-full chart: the Legendre map is regular at a state precisely when its velocity Hessian there is nonsingular. The evaluation of the Hessian for the model's mechanical presentations, and the conclusion that those presentations are regular at every state, are proved on this page (the local Legendre theorem) and are not part of this definition. If $Q=\mathrm{\varnothing }$, the Cartesian state domain is empty and regularity there is vacuous; there is no fibre-full chart instance of the cited mathematical definition and no generalized-coordinate map, whose domain $\mathrm{\Omega }$ must be nonempty.

(c) Local inverse branch data. Let the presentation be velocity-regular at a state $z$ of a chart with Legendre map $\mathbb{F}$ and momentum image $z^{\mathrm{\prime }}:=\mathbb{F}(z)$. A local inverse branch at $z$ is a pair of open sets $W\ni z$ and $W^{\mathrm{\prime }}\ni z^{\mathrm{\prime }}$ together with a $C^1$ map $v_W:W^{\mathrm{\prime }}\to W$ that is inverse to $\mathbb{F}{\mathrm{\mid }}_W:W\to W^{\mathrm{\prime }}$; equivalently, $\mathbb{F}{\mathrm{\mid }}_W$ is a $C^1$ diffeomorphism onto $W^{\mathrm{\prime }}$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. A local inverse branch assigns to every momentum covector in $W^{\mathrm{\prime }}$ a state in $W$ whose fibre derivative is that covector. Whether such a branch exists at a regular state, and when it extends to a global fibre inverse, are conditional statements of the local and global Legendre theorems of this page.

(d) Status. The item fixes the regularity vocabulary of the presentation and the data used by the local Legendre theorem; it asserts no existence of inverse branches, no hyperregularity, no equation of motion and no empirical value. The regularity condition is a condition on the declared chart data $(U,\mathsf{M})$ and, in the coordinates $X$, on the declared map $X$; no such condition is imposed on the residual forces $d$, which do not enter the velocity Hessian of $L=K-U$.

• Regularity is not hyperregularity. Regularity is a pointwise nonsingularity condition, whereas the existence of a globally defined inverse Legendre branch on a whole momentum fibre is the separate condition of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity; the two are compared on this page, and the example $\mathcal{L}(v)=e^v$ of the companion page shows that a presentation can be regular at every state and still fail to admit a global fibre inverse. The mechanical presentations of this framework lie on the positive side for both conditions, as the local and global Legendre theorems record.

• Source caveat. Tong's Section 4.1.1 phrases the invertibility requirement through convexity of the Lagrangian and the existence of a maximum; the scaffold's source audit records that this phrasing is too restrictive for local mechanical Legendre inversion, where the relevant hypothesis is nonsingularity of the velocity Hessian and a global inverse is needed only for global equivalence. The definition above follows the nonsingularity criterion; convexity is nowhere assumed, and the cited mathematical items prove the criterion.