Varying a path

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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

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Exercise 1.7. Properties of \displaystyle{\delta}

The meaning of \displaystyle{\delta_\eta (fg)[q]} is

\displaystyle{\delta_\eta (f[q]g[q])}

— Me@2019-04-27 07:02:38 PM

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2019.04.27 Saturday ACHK