Legendre transform in mechanics

For a regular Lagrangian, $p=\partial L/\partial \dot{q}$ and $H=p\dot{q}-L$. Regularity means the velocity Hessian is invertible.

A Lagrangian is regular when its velocity Hessian matrix is invertible. The inverse function theorem then makes the fiber derivative $p_i=L_{v^i}$ locally invertible, allowing a local Legendre transform in mechanics to a Hamiltonian. If that fiber derivative is globally invertible, the Lagrangian is called a hyperregular Lagrangian. Regularity alone does not assert global invertibility, and singular Lagrangians may give constrained rather than ordinary unconstrained Hamilton's equations.

A Lagrangian is hyperregular when its fiber derivative, sending a velocity to its conjugate momentum, is a global diffeomorphism from the tangent bundle to the cotangent bundle. This gives a globally defined Legendre transform in mechanics and a global equivalence between the Euler-Lagrange equations and Hamilton's equations. With time dependence this requirement is imposed at each time, with a smooth inverse depending on time.

For $L=mc|\dot{x}|-V(x)$,The momentum loses the speed, so the velocity cannot be recovered from $(x,p)$ and the ordinary inverse Legendre transform does not exist.