Understanding an application of the chain rule to partial derivatives (Lagrangian setting)

Question

On pg. 97 of No-Nonsense Classical Mechanics, the author uses the chain rule as follows:

I am totally confused as to why the result of the chain rule applied to

$$ \frac{\partial L \left( q, \dot{q}(q, p) \right)}{\partial q} $$

yields a sum:

$$ \frac{\partial L ( q, \dot{q} ) }{\partial q} + \frac{\partial L(q, \dot{q}) }{\partial \dot{q}} \frac{ \partial \dot{q} (q, p) }{ \partial q} $$

Where does the sum come from?

Answer 1

You sum:

• the direct dependency of $L$ on $q$
• the indirect dependency of $L$ on $q$ through $\dot{q}$.

That's just the chain rule :)

Note: the notation of the total derivative should be $\frac{dL}{dq}$.