# 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