Regular vs stable orbits in spherically symmetric potentials

[ZelchJ]

TL;DR Summary

Are chaotic orbits possible in central force fields?

I am struggling with Hamiltonian formulation of classical mechanics. I think I have grasped the idea of canonical transformations, including the idea of angle-action variables and invariant tori in phase space. Still, few points seem to elude my understanding...

Let's talk about a particle moving in a 3D potential, so its phase-space is 6D. If I understand it right, if this potential has at least 3 integrals of motion, then the system is said to be integrable, and the particle's orbit in phase space is confined to the surface of a three-dimensional torus, and its motion in Cartesian coordinates is quasiperiodic. If, on the other hand, there are less than 3 integrals of motion, then some (if not all) orbits are chaotic (irregular).

Now, any static spherically symmetric potential has four integrals of motion - energy and three components of the angular momentum. Is this enough to state that chaotic orbits are impossible in static spherically symmetric potentials? In other words, that all orbits are regular in any static central force field?

If yes, then how does this match with the fact that unstable orbits are possible in central force field? (e.g., https://physics.stackexchange.com/questions/183726/what-makes-an-orbit-stable-or-unstable). In my understanding, regular orbits are basically a synonym for stable orbits: because regular orbits are not chaotic, two near-by orbits on two near-by tori don't ever diverge as chaotic orbits do. Isn't it the same as stability?

 

[DrClaude]

If yes, then how does this match with the fact that unstable orbits are possible in central force field? (e.g., https://physics.stackexchange.com/questions/183726/what-makes-an-orbit-stable-or-unstable). In my understanding, regular orbits are basically a synonym for stable orbits: because regular orbits are not chaotic, two near-by orbits on two near-by tori don't ever diverge as chaotic orbits do. Isn't it the same as stability?

Stability here means orbits that are closed, i.e., that do not converge (diverge) to 0 (infinity).

 

[jbriggs444]

Suppose that one uses a potential that has a local maximum. Further, assume that the potential is shaped so that a particle can approach the point of the maximum and arrive within finite time with zero remaining kinetic energy. Then the laws of physics are impredictive about what happens next. The particle can stay there indefinitely. Or it can depart at any time in either direction.

If my memory does not betray me, the function ##f(x) = -e^{-1/x^2}## for x not equal to zero and ##f(x) = 0## for ##x = 0## qualifies as such a potential. One could tweak it to put a potential barrier at the ends so that escape is made impossible.

 

[vanhees71]

Of course, for this potential the analysis in terms of a power series around ##x=0## must fail, because there is an essential singularity when considered as a complex function, and thus there's no power series or, more precisely, the power series has convergence radius 0.

 

[ZelchJ]

Thank you all for the replies. But I still can't see how this relates to the notion of regular orbits in phase-space of Hamiltonian formulation.