Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian

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 $Q\subseteq {\mathbb{R}}^{3N}$, 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 $U:I_0\times Q\to \mathbb{R}$ 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 $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$, kinetic metric $G(q)=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q)$ and substituted potential $\tilde{U}(t,q)=U(t,X(q))$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces), and let $H$ and $\tilde{H}$ be the Hamiltonians of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions. Then:

1. (Cartesian chart) The fibre momentum map ${\mathbb{F}}_{t,x}(v)=\mathsf{M}v$ is a bijection of ${\mathbb{R}}^{3N}$ onto ${\mathbb{R}}^{3N}$ for every $(t,x)\in I_0\times Q$, and the inverse Legendre branch is the $C^2$ map $v(t,x,p)={\mathsf{M}}^{-1}p$ on the whole phase chart $I_0\times Q\times {\mathbb{R}}^{3N}$. Hence the presentation is hyperregular on the Cartesian chart in the sense of Hyperregular potential presentations and their inverse Legendre branches in the model's charts, and the Hamiltonian of the branch is

$H(t,x,p)=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U(t,x),$

with the dot product of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$,

a $C^2$ function of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.

2. (generalized-coordinate chart) The fibre map $v\mapsto G(q)v$ is a bijection of ${\mathbb{R}}^r$ onto ${\mathbb{R}}^r$ for every $(t,q)\in I_0\times \mathrm{\Omega }$, and the inverse Legendre branch is the $C^2$ map $v(t,q,p)=G(q{)}^{-1}p$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$. Hence the presentation is hyperregular on the chart of $X$, and the Hamiltonian in the coordinates $X$ is

$\tilde{H}(t,q,p)=\frac{1}{2}p\cdot G(q{)}^{-1}p+\tilde{U}(t,q),$

a $C^2$ function of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.

The statement is conditional on the adopted force branch and the declared presentation; it asserts no equation of motion, no conservation law and no empirical content.

• The model is never singular. Clauses 1 and 2 show that the mechanical presentations of the framework are hyperregular on every chart of the model: the positive definiteness of the mass operator and of the kinetic metric makes each fibre momentum map a linear isomorphism, so the inverse branch is global in the fibre and given in closed form. This is the positive case anticipated by Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts and Hyperregular potential presentations and their inverse Legendre branches in the model's charts; no such conclusion holds for general Lagrangians, as the companion page's examples record.

• A linear specialisation of the cited proposition. The mathematical content is the proposition A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian instantiated at the model's data; the physical content is the identification of those data with the declared presentation, which fixes the meanings $p=\mathsf{M}v$, $v={\mathsf{M}}^{-1}p$ and $H=\frac{1}{2}p\cdot {\mathsf{M}}^{-1}p+U$ in terms of the labels, masses and potential. The residual forces $d$ do not enter the branch or the Hamiltonian; they reappear as momentum sources in the equivalence theorem of this page.