Hyperregular potential presentations and their inverse Legendre branches in the model's 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 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 $U$ 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 differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Cartesian chart. The presentation is hyperregular on the Cartesian chart when for every $(t,x)\in I_0\times Q$ the fibre momentum map

${\mathbb{F}}_{t,x}:{\mathbb{R}}^{3N}⟶{\mathbb{R}}^{3N},{\mathbb{F}}_{t,x}(v):=L_v(t,x,v)=\mathsf{M}v,$

is a bijection of ${\mathbb{R}}^{3N}$ onto ${\mathbb{R}}^{3N}$ and the inverse assignment

$v:I_0\times Q\times {\mathbb{R}}^{3N}⟶{\mathbb{R}}^{3N},L_v(t,x,v(t,x,p))=p,$

is defined on the whole momentum phase chart $I_0\times Q\times {\mathbb{R}}^{3N}$ and is of class $C^1$. In that case $v$ is the inverse Legendre branch of the presentation on the Cartesian chart. For $Q\mathrm{\ne }\mathrm{\varnothing }$, hyperregularity of this chart is exactly hyperregularity of its fibre-full Lagrangian in the sense of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, read through the identification of the fibre derivative with the momentum of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta. If $Q=\mathrm{\varnothing }$, the fibre condition is vacuous and the inverse assignment is the unique empty $C^1$ map; this is an empty-domain convention, not an instance of that supplier's fibre-full chart, whose base must be nonempty.

(b) Generalized-coordinate chart. Let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate map with substituted Lagrangian $\tilde{L}$ and kinetic metric $G$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces). The presentation is hyperregular on the chart of $X$ when for every $(t,q)\in I_0\times \mathrm{\Omega }$ the fibre map $v\mapsto {\tilde{L}}_v(t,q,v)=G(q)v$ is a bijection of ${\mathbb{R}}^r$ onto ${\mathbb{R}}^r$ and the inverse assignment $v(t,q,p)$ on $I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r$ is of class $C^1$; $v$ is then the inverse Legendre branch in the coordinates $X$. Linear maps and bijectivity are read by Linear map between vector spaces over the same field, and the pairing of momentum covectors with velocities is the Euclidean pairing of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$.

(c) Status. The item defines the hyperregularity condition and the branch objects for the two chart types of the model. It asserts no branch existence, no relation between regularity Velocity-regular potential presentations: the velocity Hessian of the mechanical Lagrangian, regular states and regular charts and hyperregularity, no equation of motion and no empirical value. The global Legendre theorem for the mechanical branch on this page establishes hyperregularity for every admitted Cartesian and generalized-coordinate presentation. In the broader class of fibre-full $C^2$ Lagrangian charts of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, regularity alone does not imply hyperregularity: the companion page's example $\mathcal{L}(v)=e^v$ has a nonsurjective fibre momentum map. That example is outside the model's mechanical class and witnesses the separation only in this broader class.

• Two degrees of globality. Hyperregularity has two quantifiers: the fibre map must be bijective for every $(t,q)$ (surjectivity and injectivity on each momentum fibre), and the inverse assignment must be defined on the whole phase domain and be $C^1$ there. Injectivity alone gives no description of which momenta occur. Fibrewise bijectivity determines a unique inverse assignment; its joint $C^1$ regularity on the whole phase domain is a further requirement of the definition.

• The inverse branch is attached to the declared chart. The next definition of this page defines the Hamiltonian of the presentation from the inverse Legendre branch introduced here; consequently each Hamiltonian statement names the chart on which hyperregularity and the branch are taken, and on a fixed hyperregular chart the inverse is unique by fibre bijectivity. Different charts express the inverse in their own coordinates. The later global Legendre theorem on this page proves its existence for every admitted mechanical presentation; that conclusion is not an existence assertion of the present definition.

• Source caveat. The cited treatments invert the velocity-momentum relation on the branch where it is invertible and do not distinguish the pointwise, local and global levels of the inverse; the scaffold's source audit notes that a global inverse is needed for global equivalence between the Lagrangian and Hamiltonian descriptions, which is why the condition above is stated globally in the fibre. The mathematical content of regularity, local inversion and global inversion is supplied by the cited items of this page, not by the physical treatments.