8.3: Varying Both Ends

The argument given above for the incremental change in action from varying the endpoint is clearly equally valid for varying the beginning point of the integral (there will be a sign change, of course), so

$$\begin{equation} d S\left(q_{i}^{(2)}, t_{2}, q_{i}^{(1)}, \quad t_{1}\right)=\sum_{i} p_{i}^{(2)} d q_{i}^{(2)}-H^{(2)} d t_{2}-\sum_{i} p_{i}^{(1)} d q_{i}^{(1)}+H^{(1)} d t_{1} \end{equation}$$

The initial and final coordinates and times specify the action and the time development of the system uniquely.

(Note: We’ll find this equation again in the section on canonical transformations -- the action will be seen there to be the generating function of the time-development canonical transformation, this will become clear when we get to it.)