Hyperregular Lagrangian ($\mathbb{F}L:TQ\stackrel{\sim }{⟶}{T}^{\ast }Q$)

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.