Scope of the Legendre map: regular versus hyperregular, the meaning of the covector datum, and the limits of the branch description

Regularity is local; hyperregularity is global in the fibre. The derivative criterion of Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse produces inverse branches only near points where the velocity Hessian is nonsingular, whereas hyperregularity as defined in The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity demands a bijection of each velocity fibre onto the full momentum fibre with globally defined inverse. Hyperregularity implies regularity: differentiating the inverse fibre identity gives $D_{p}v{\mathcal{L}}_{vv}=I_n$, hence a nonsingular Hessian. Regularity does not imply hyperregularity, as the bounded-gradient example shows; and the propositions on this page state explicitly which hypothesis each conclusion uses: the differential and energy identities The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity use a branch, and the elimination in Stationarity of the phase-space action is equivalent to Hamilton's equations, and hyperregular elimination recovers the Lagrangian action uses a global branch for the chart.

The mechanical chart is the positive example. For $\mathcal{L}(t,q,v)=\frac{1}{2}{v}^{\mathsf{T}}G(t,q)v+A(t,q)\cdot v-V(t,q)$ with $G$ symmetric positive definite, A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian proves hyperregularity and computes the Hamiltonian explicitly. The companion page records the degenerate cases: a regular Lagrangian whose velocity gradient is not onto the momentum fibre is not hyperregular, and the identically zero Lagrangian is not even regular, so it admits no Legendre inverse at all.

The covector datum $A$ and the momentum. In the mechanical chart the momentum is $p=Gv+A$; it is a covector determined by the chart data, not $\text{mass}\times \text{velocity}$ unless $G$ is a constant scalar matrix and $A=0$. The datum $A$ is a mathematical field on the chart and is not, in this development, an electromagnetic potential; identifying $A$ with a physical vector potential is a modelling step belonging to the consuming physics items, and requires its own assumptions and units.

No completeness or global solvability is asserted. The branch description of the motion is local in the momentum variable and requires the invertibility hypotheses above. It does not assert that the Euler–Lagrange or Hamilton equations have global solutions, and the finite-time breakdown of a solution is not excluded by anything on this page.