Chain rule of functional variation

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Ex 1.8.2.3, Structure and Interpretation of Classical Mechanics

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\displaystyle{ \begin{aligned} &\delta_\eta F[g[q]] \\ &= \delta_\eta (F \circ g)[q] \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q + \epsilon \eta]] - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q] + \epsilon \delta_\eta g[q] + \epsilon^2 (...) + \epsilon^3 (...) + ...]] - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q] + \epsilon \delta_\eta g[q] + \epsilon^2 (... + \epsilon (...) + ...)]] - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q] + \epsilon \delta_\eta g[q] + \epsilon^2 (...)]] - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q] + \epsilon \left(\delta_\eta g[q] + \epsilon (...)\right)]] - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{F[g[q]] + \epsilon \delta_{\left(\delta_\eta g[q] + \epsilon (...)\right)} F[g[q]] + \epsilon^2 (...) - F[g[q]]}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \frac{\epsilon \delta_{\left(\delta_\eta g[q] + \epsilon (...)\right)} F[g[q]] + \epsilon^2 (...)}{\epsilon} \right) \\ &= \lim_{\epsilon \to 0} \left( \delta_{\left(\delta_\eta g[q] + \epsilon (...)\right)} F[g[q]] + \epsilon (...) \right) \\ &= \delta_{ \left( \delta_\eta g[q] \right)} F[g] \\ \end{aligned}}

— Me@2020-07-14 06:00:35 PM

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2020.07.15 Wednesday (c) All rights reserved by ACHK