I'm studying action-angle coordinates and I've come across two distinct, yet correct, definitions for the new generalized momenta, $\text{J}_k$:

$$\text{J}_k \equiv \oint p_k dq_k $$


$$\text{J}_k \equiv \frac{1}{2\pi} \oint p_k dq_k .$$

I'm trying to understand how they relate.

  • 2
    $\begingroup$ one includes a factor of $2\pi$ and the other one does not? it is just about conventions, there is no physics behind that factor (besides the fact that it simplifies the harmonic oscillator) $\endgroup$ – AccidentalFourierTransform Nov 28 '16 at 17:40
  • $\begingroup$ Doesn't it have to do with normalization or something? I'm trying to understand why one has the factor and the other does not. $\endgroup$ – Phy09 Nov 28 '16 at 17:42
  • $\begingroup$ So, I can use either one for any case and I'll obtain the same frequencies, same equations of motion etc? But wouldn't the new generalized momenta, $\text{J}_k$, differ by a factor of $2\pi$ depending on which definition I used? $\endgroup$ – Phy09 Nov 28 '16 at 17:46

Imagine you have the harmonic oscillator

$$ H(q, p) = \frac{p^2}{2} + \frac{q^2}{2} \tag{1} $$

In phase space, the are enclosed by a solution of Eq. (1) is

$$ A = \pi\times\sqrt{2}\times \sqrt{2} = 2\pi $$

For this orbit, the action is

$$ J = \frac{1}{2\pi}\oint {\rm d}q\; p = \frac{1}{2\pi}A = 1 $$

That is the reason behind the "$2\pi$", but it is absolutely not required. The action $J$ will have the same meaning even if this factor is removed


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