In Chapter 8, pages 86-87, equations (8.5)-(8.11) of Julian Schwinger et al., Classical Electrodynamics, the equations of motion for the following action principle of a point particle in an external potential are derived in an unconventional manner, from what I've seen.
The action is given as:
$$ W_{12} = \int_2^1 \left[\frac{1}{2}m\frac{(\mathrm d\bf r)^2}{\mathrm dt} - \mathrm dt\, V(\mathbf r,t)\right]\tag{8.10} $$
with the variations on $$\mathbf r(t) \to \mathbf r(t) + \delta\mathbf r(t)\tag{8.5}$$ and $$ t \to t + \delta t(t),\tag{8.6}$$ such that at the endpoints $$\delta t(t_1) = \delta t_1, \qquad \delta t(t_2) = \delta t_2,\tag{8.7}$$ so that the limits of integration are not changed. Thus time is a function of the parameter time $t$ (what I assume is notational abuse). This is different from the usual approach in which $\mathbf r$ and $\dot{\mathbf r}$ are considered as independent variables and varied accordingly.
The corresponding changes in the time differential and time derivative are:
$$ \mathrm dt \to d(t+\delta t) = \left(1 + \frac{\mathrm d\delta t}{\mathrm d t}\right)\mathrm dt,\tag{8.8}$$ $$ \frac{\mathrm d}{\mathrm dt} \to \left(1 - \frac{\mathrm d \delta t}{\mathrm d t}\right)\frac{\mathrm d}{\mathrm dt}.\tag{8.9}$$
The following variation is then presented:
$$\delta W_{12} = \int^1_2 \mathrm dt\left\{m\frac{\mathrm d\bf r}{\mathrm dt}\cdot\frac{\mathrm d}{\mathrm dt}\delta\mathbf r - \delta\mathbf r \cdot \nabla V - \frac{\mathrm d\delta t}{\mathrm dt}\left[\frac{1}{2}m\left(\frac{\mathrm d\bf r}{\mathrm dt}\right)^2 + V \right] - \delta t \frac{\partial}{\partial t}V\right\}. \tag{8.11}$$
I've attempted to figure out how to derive this, but keep getting stuck at the first step itself. Here are some possibilities I considered:
- Start with the variation of the action defined as an integral over the Lagrangian and vary sensibly:
$$ \delta W_{12} = \delta \left(\int_2^1 L(\mathbf r, \dot{\mathbf r}, t)\right) = \int^1_2 \left[\delta L\,\mathrm dt + L\,\delta(\mathrm dt)\right] $$
- Treat variation as a "transformation", i.e. substitute the transformed parameters in the Lagrangian:
$$ \int_2^1 L\left[\mathbf r + \delta\mathbf r, \left(1 - \frac{\mathrm d\delta t}{\mathrm dt}\right)\frac{\mathrm d}{\mathrm dt}\left(\mathbf r + \delta\mathbf r(t)\right), t + \delta t\right]\left[1 + \frac{\mathrm d\delta t}{\mathrm dt}\right]\mathrm dt $$
- Consider the variation of the free particle's kinetic energy term, which leads to the following expression as I see it:
\begin{align*} \delta\left(\frac{\mathrm d\mathbf r}{\mathrm dt}\right)^2 & = 2\left(\frac{\mathrm d\mathbf r}{\mathrm dt}\right)\left[\delta\left(\frac{\mathrm d}{\mathrm dt}\right)\mathbf r + \frac{\mathrm d}{\mathrm dt}\delta\mathbf r\right] \\ & = 2\left(\frac{\mathrm d\mathbf r}{\mathrm dt}\right)\left[\left(1-\frac{\mathrm d\delta t}{\mathrm dt}\right)\frac{\mathrm d\mathbf r}{\mathrm dt} + \frac{\mathrm d}{\mathrm dt}\delta\mathbf r\right] \end{align*}
After the usual Taylor expansion up to first order, I can't see any of these leading to the variation given in the book. Which method, if any, is correct? I'm also not particularly sure if 3. is correct.