A Tale of Two L’s

Lagrange’s equations are traditionally written in the form

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}} \right ) = \frac {\partial L}{\partial q}}

or, if we write a separate equation for each component of \displaystyle{q}, as

\displaystyle{\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q^i}} \right ) = \frac {\partial L}{\partial q^i}}

In this way of writing Lagrange’s equations the notation does not distinguish between \displaystyle{L}, which is a real-valued function of three variables \displaystyle{(t, q, \dot q)}, and \displaystyle{L \circ \Gamma[q]}, which is a real-valued function of one real variable \displaystyle{t}.

— Structure and Interpretation of Classical Mechanics

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2019.09.04 Wednesday ACHK