8.3: Varying Both Ends

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Page ID: 29578

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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.)