Understanding the SI units of the Euler–Lagrange equation terms

Page 88 of No-Nonsense Classical Mechanics states the Euler-Lagrange equation as follows: Questions: I’m having trouble understanding just what the statement means.

1. Are $$frac{partial L}{partial q}$$ and $$frac{partial L}{partial dot{q}}$$ directional derivatives?

2. What are the SI units of $$frac{partial L}{partial q}$$, and won’t they differ from $$frac{d}{dt} left( frac{partial L}{partial q} right)$$ due to the $$frac{d}{dt}$$? If the units differ between these two terms, doesn’t this equation "fail to make sense"?

Mathematics Asked by user1770201 on December 29, 2020

Let's suppose the units of $$q$$ are are $$text{u}$$, which stands for "user units."

The units of $$frac{partial L}{partial q}$$ are $$text{J}text{u}^{-1}$$ (Joules per user unit.)

The units of $$frac{partial L}{partial dot q}$$ are Joules per (user units per second) which is $$text{J}text{u}^{-1}text{s}$$. Hence the units of $$frac{d}{dt}left(frac{partial L}{partial dot q}right)$$ are $$text{J}text{u}^{-1}$$.

You could think of them as directional derivatives, but I think it is more helpful just to think of them as partial derivatives.

Correct answer by Stephen Montgomery-Smith on December 29, 2020

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