Varying a path

Suppose that we have a function $\displaystyle{f[q]}$ that depends on a path $\displaystyle{q}$ . How does the function vary as the path is varied? Let $\displaystyle{q}$ be a coordinate path and $\displaystyle{q + \epsilon \eta}$ be a varied path, where the function $\displaystyle{\eta}$ is a path-like function that can be added to the path $\displaystyle{q}$ , and the factor $\displaystyle{\epsilon}$ is a scale factor. We define the variation $\displaystyle{ \delta_\eta f[q]}$ of the function $\displaystyle{f}$ on the path $\displaystyle{q}$ by

$\displaystyle{\delta_\eta f [q] = \lim_{\epsilon \to 0} \left( \frac{f[q + \epsilon \eta] - f[q]}{\epsilon} \right)}$

The variation of $\displaystyle{f}$ is a linear approximation to the change in the function $\displaystyle{f}$ for small variations in the path. The variation of $\displaystyle{f}$ depends on $\displaystyle{\eta}$ .

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

Exercise 1.7. Properties of $\displaystyle{\delta}$