Lagrange’s equations Debugged

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\displaystyle{\frac{d}{dt} \left( \frac{\partial L(t, q, \dot q)}{\partial \dot q} \Bigg|_{\begin{aligned} q &= w(t) \\ \dot q &= \frac{d w(t)}{dt} \\ \end{aligned}} \right) - \frac{\partial L}{\partial q}\Bigg|_{\begin{aligned} q &= w(t) \\ \dot q &= \frac{d w(t)}{dt} \\ \end{aligned}} = 0}

This equation is complete. It has meaning independent of the context and there is nothing left to the imagination. The earlier equations require the reader to fill in lots of detail that is implicit in the context. They do not have a clear meaning independent of the context.

\displaystyle{\frac{d}{dt} \frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} = 0}

— Functional Differential Geometry

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2023.09.05 Tuesday ACHK