A system with one degree of freedom has Lagrangian $L(q, \dot{q})$. Define the canonical momentum $p$ and the energy $E$. Show that $E$ is constant along any classical path.

Consider a classical path $q_{c}(t)$ with the boundary-value data

$$q_{c}(0)=q_{I}, \quad q_{c}(T)=q_{F}, \quad T>0$$

Define the action $S_{c}\left(q_{I}, q_{F}, T\right)$ of the path. Show that the total derivative $d S_{c} / d T$ along the classical path obeys

$$\frac{d S_{c}}{d T}=L$$

Using Lagrange's equations, or otherwise, deduce that

$$\frac{\partial S_{c}}{\partial q_{F}}=p_{F}, \quad \frac{\partial S_{c}}{\partial T}=-E,$$

where $p_{F}$ is the final momentum.