Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 9D (Classical Dynamics)

For an autonomous Lagrangian, the canonical momentum and energy areAlong a solution of the Euler-Lagrange equation,The absence of explicit time dependence is essential to this conservation law.

The classical action is $S_c(q_I,q_F,T)={\int }_0^{T}L(q_c,{\dot{q}}_c)dt$. If the final endpoint is moved along one fixed classical trajectory, while the initial endpoint is held fixed, the fundamental theorem of calculus givesThis total derivative is different from the partial derivative with fixed final position.

For a family of nearby classical trajectories, integration by parts gives the endpoint variationTo obtain the final term, the endpoint change in the variation at fixed parameter time is $\delta q(T)=\delta q_F-{\dot{q}}_F\delta T$, while the changing upper integration limit contributes $L_F\delta T$. The interior integral vanishes by the Euler-Lagrange equation. Therefore, on any smooth branch of classical boundary-value solutions,The chain rule along the endpoint trajectory now reads $d{S}_c/dT=p_F{\dot{q}}_F-E=L_F$, consistently with the direct calculation.