 Energy balance for an autonomous Lagrangian

For an autonomous Lagrangian $L(x,\dot{x})$, define its action $A=\int Ldt$ and Hamiltonian $H=\dot{x}{L}_{\dot{x}}-L$. The chain rule givesThus the identity holds along forced trajectories as well as solutions of the homogeneous Euler-Lagrange equation. Explicit time dependence adds $-{\partial }_{t}L$; stochastic work requires a consistent regularization or the Stratonovich chain rule.