8.4: Another Way of Writing the Action Integral

Up to this point, we’ve always written the action as an integral of the Lagrangian with respect to time along the path,

$$\begin{equation} S\left(q_{i}^{(2)}, t_{2}, q_{i}^{(1)}, \quad t_{1}\right)=\int_{q^{(1)}, t_{1}}^{q^{(2)}, t_{2}} L d t \end{equation}$$

However, the expression derived in the last section for the increment of action generated by an incremental change in the path endpoint is clearly equally valid for the contribution to the action from some interior increment of the path, say from $\begin{equation} (q, t) \text { to }(q+d q, t+d t) \end{equation}$ so we can write the total action integral as the sum of these increments:

$$\begin{equation} S\left(q_{i}, t\right)=\int d S=\int\left(\sum_{i} p_{i} d q_{i}-H d t\right) \end{equation}$$

In this integral, of course, the $\begin{equation} d q_{i} \end{equation}$ add up to cover the whole path.

(In writing $\begin{equation} \left(q_{i}, t\right) \end{equation}$ we’re following Landau’s default practice of taking the action as a function of the final endpoint coordinates and time, assuming the beginning point to be fixed. This is almost always fine—we’ll make clear when it isn’t.)