1

According to Goldstein,

"We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (libration or periodic functions of $q_i$)." (p. 457, 3rd Ed.)

I'm working through the following example: for a particle of mass $m$ in a generic central potential $V = kr^\alpha$, let us fix the total energy ($E$) and angular momentum ($L$) of the system. We would then have the following expression for the radial momentum: $$ p_r (r) = \sqrt{2m\bigg(E - kr^\alpha - \frac{L}{2mr^2}\bigg)}\ . $$ Is it correct then, that to determine the conditions under which action-angle variables for this system exist, I would have to set $p_r(r) = p_r(r + a)$, where $a$ would denote the period, and derive constraints on the constants $E, L, k$, and $\alpha$?

Qmechanic
  • 201,751
Anthony
  • 173
  • related to https://physics.stackexchange.com/q/206570/ – ZeroTheHero Sep 02 '18 at 15:10
  • The answer to that question doesn't really help me, since the "loop integral" definition is precisely what I'm trying to understand--what is the period of this loop? In particular, he substitutes in the known solution of $q$ as a function of angle and integrates around its period $2 \pi$, obtaining nothing more than an a posteriori confirmation of the solution. – Anthony Sep 02 '18 at 16:14
  • The circuit integral is between the limits of the motion. – ZeroTheHero Sep 02 '18 at 17:26
  • @ZeroTheHero Ok, but does this guarantee that the motion in phase space will be periodic? That seems to be a precondition to defining the action-angle variables. – Anthony Sep 03 '18 at 14:58

0 Answers0