 Solution

Stationarity of the action under variations with fixed endpoints gives the Euler-Lagrange equation. The generalized momentum is $p=\partial L/\partial \dot{q}$. The Legendre transform in mechanics isafter solving $p(q,\dot{q},t)$ for $\dot{q}$. This requires at least local invertibility, ${\partial }^{2}L/\partial {\dot{q}}^2\ne 0$; global single-valuedness requires global invertibility.