Regular, hyperregular and almost-regular Legendre domains

Assume countable choice ACω, as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), for the smooth cotangent-bundle carrier convention used by this page; no full Axiom of Choice is assumed.

Let $\mathcal{D}$ be an open subset of the time-dependent tangent bundle $\mathbb{R}\times TQ$ of a finite-dimensional smooth manifold $Q$, and let $L:\mathcal{D}\to \mathbb{R}$ be smooth. For fixed $t$, the fibre derivative is the map

$\mathbb{F}L(t,q,v)=(t,q,p),p(w)={\frac{d}{ds}\mid }_{s=0}L(t,q,v+sw),$

as in Fibre derivative or Legendre map of a Lagrangian. In induced coordinates $p_i=\mathrm{\partial }L\mathrm{/}\mathrm{\partial }{v}^i$.

The Lagrangian is regular at $(t,q,v)$ when the velocity Hessian $({\mathrm{\partial }}^{2}L\mathrm{/}\mathrm{\partial }{v}^i\mathrm{\partial }{v}^j)$ is invertible there; regularity gives only a local inverse branch. It is hyperregular on a declared domain when $\mathbb{F}L$ is a diffeomorphism from that entire domain onto an open subset of $\mathbb{R}\times T^{\ast }Q$.

It is almost regular on a declared open domain ${\mathcal{D}}_0\subseteq \mathcal{D}$, with the inherited open-submanifold structure, when the following are all assumed: $\mathbb{F}L{\mathrm{\mid }}_{{\mathcal{D}}_0}$ has constant rank; its image $N$ is an embedded submanifold of $\mathbb{R}\times T^{\ast }Q$; its corestriction ${\mathcal{D}}_0\to N$ is a surjective submersion (its differential is surjective at every point); and every fibre of that corestriction is connected. These hypotheses are not consequences of singularity alone.