The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity

Chart data. Let $n\ge 1$, let $J\subseteq \mathbb{R}$ and $U\subseteq {\mathbb{R}}^n$ be open and nonempty, and let

$\mathcal{L}:J\times U\times {\mathbb{R}}^n⟶\mathbb{R}$

be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms). Such a map is a fibre-full time-dependent Lagrangian chart; it is the special case of the chart of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations in which the velocity block is the whole space ${\mathbb{R}}^n$ at every $(t,q)\in J\times U$. Write ${\mathcal{L}}_v(t,q,v)\in {\mathbb{R}}^n$ for the Euclidean gradient of $\mathcal{L}$ in its third slot and ${\mathcal{L}}_{vv}(t,q,v)\in M_n(\mathbb{R})$ for the matrix of second partial derivatives in the third slot (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

Legendre map. The Legendre map of $\mathcal{L}$ is the map

${\mathbb{F}}_{\mathcal{L}}:J\times U\times {\mathbb{R}}^n⟶J\times U\times {\mathbb{R}}^n,{\mathbb{F}}_{\mathcal{L}}(t,q,v):=(t,q,{\mathcal{L}}_v(t,q,v))\mathrm{.}$

The fibre coordinate of the target is called the momentum and is written $p:={\mathcal{L}}_v(t,q,v)$. Thus ${\mathbb{F}}_{\mathcal{L}}$ is the identity in the first two slots and the gradient in the velocity slot. Since $\mathcal{L}$ is $C^2$, ${\mathbb{F}}_{\mathcal{L}}$ is $C^1$ ($C^k$ Euclidean maps and diffeomorphisms).

Regularity. Let $(t,q,v)\in J\times U\times {\mathbb{R}}^n$. The Legendre map is regular at $(t,q,v)$ when the velocity Hessian is nonsingular,

$\det {\mathcal{L}}_{vv}(t,q,v)\mathrm{\ne }0;$

it is regular (on the chart) when it is regular at every point. At fixed $(t,q)$ this is exactly the condition under which the map $v\mapsto (q,{\mathcal{L}}_v(t,q,v))$ is locally invertible near $v$, a statement made precise and proved in Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse.

Hyperregularity. The Lagrangian $\mathcal{L}$ is hyperregular (on the chart) when for every $(t,q)\in J\times U$ the map

$v⟼{\mathcal{L}}_v(t,q,v)$

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

$(t,q,p)⟼v(t,q,p),{\mathcal{L}}_v(t,q,v(t,q,p))=p,$

is defined on all of $J\times U\times {\mathbb{R}}^n$ and is of class $C^1$. In that case $v$ is called the inverse Legendre branch of $\mathcal{L}$, and the Hamiltonian of the branch is

$H:J\times U\times {\mathbb{R}}^n⟶\mathbb{R},H(t,q,p):=p\cdot v(t,q,p)-\mathcal{L}(t,q,v(t,q,p))\mathrm{.}$

The Hamiltonian is $C^1$ on its domain, being a sum of products of $C^1$ maps.

The definition asserts nothing about solving the equations of motion: the equivalence of the Euler–Lagrange equation with Hamilton's equations on an inverse branch, and the differential of $H$, are proved in The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity.

Scope and warnings. Hyperregularity is a global-in-the-fibre hypothesis: it requires the velocity Legendre map to be bijective onto the whole momentum fibre ${\mathbb{R}}^n$, with globally defined inverse. Regularity is a local condition. Hyperregularity implies regularity: at fixed $(t,q)$, differentiating $v(t,q,{\mathcal{L}}_v(t,q,w))=w$ gives $D_{p}v{\mathcal{L}}_{vv}=I_n$, so the square velocity Hessian is invertible. The converse fails: a regular Lagrangian whose velocity gradient is not surjective is not hyperregular, and a fibre-full domain whose Legendre map fails to be injective has no inverse branch. Concrete witnesses — including the mechanical Lagrangian of A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian, the bounded gradient $\mathcal{L}(v)=\sqrt{1+v^2}$ and the identically zero Lagrangian — are collected in Regular versus hyperregular: the bounded-gradient Lagrangian $\sqrt{1+v^2}$ and the degenerate zero Lagrangian ↗ and Scope of the Legendre map: regular versus hyperregular, the meaning of the covector datum, and the limits of the branch description.

• The published bundle version of this notion — a smooth Lagrangian on a tangent bundle, its fibre derivative, and the definitions of regular and hyperregular Lagrangians by local diffeomorphism, respectively global fibre-preserving diffeomorphism, of the Legendre map — is the autonomous manifold formulation of the same concept. The present definition is the finite-dimensional chart version with explicit time dependence, which is the form consumed by the classical-mechanics items on this page; it is not a restatement of the bundle definition, because time-dependent Lagrangians and the explicit $C^1$ inverse branch on $J\times U\times {\mathbb{R}}^n$ have no counterpart in the autonomous formulation.

• When $\mathcal{L}$ is hyperregular, the fibre condition and the open domain $J\times U$ make $H$ a globally defined $C^1$ function. If only regularity holds, an inverse branch exists only locally, and the local Hamiltonian is correspondingly local; the items below state which hypothesis each conclusion uses.