Past exam of the classical mechanics course of Part B of the physics course of the University of Oxford / 2016 / 3 / a / Solution

For an action functional $S[q]={\int }_{t_i}^{t_f}L(q,\dot{q},t)dt$, vary a path while fixing its endpoint coordinates. Integration by parts givesThe endpoint term vanishes. Since the interior variations are arbitrary, stationarity givesDefine canonical momentum $p_i=\partial L/\partial {\dot{q}}_i$. For a regular velocity Hessian, these relations can be inverted for $\dot{q}(q,p,t)$, and the Legendre transform definesIts differential is $dH={\sum }_i({\dot{q}}_{i}d{p}_i-L_{q_i}d{q}_i)-L_{t}dt$. Inserting the Euler-Lagrange equations already suggests the canonical equations.

To derive them directly from an principle of stationary action, vary the first-order action $I[q,p]=\int ({\sum }_i{p}_i{\dot{q}}_i-H)dt$ with independent $q,p$ and fixed endpoint $q$. Integrating $p_i\delta {\dot{q}}_i$ by parts givesThereforeThe endpoint momenta need not be fixed. A singular velocity Hessian instead requires a constrained Hamiltonian formulation; the ordinary Legendre inversion is an assumption, not an identity valid for every Lagrangian.