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Understanding the SI units of the Euler–Lagrange equation terms

Page 88 of No-Nonsense Classical Mechanics states the Euler-Lagrange equation as follows:

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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

1 Answers

One Answer

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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