# Apparent emergence of conserved quantities in non-integrable systems

This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable?

Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities. For example, let's assume that the number of degrees of freedom is $D=3$ and that the only conserved quantity is the Hamiltonian $H$ itself. In whatever constant-energy surface of the type $H=E_0$, there will be both chaotic and regular regions. Consider a regular trajectory in phase space. The relevant motion consists in regular oscillations, so it should be possible to conveniently introduce action / angle variables.

1) Am I wrong?

Variables of the type "action" are conserved quantities.

2) Isn't it strange? Where do they come from? It seems to me that a non-integrable system can feature regular regimes where additional conserved quantities are present. Even stranger, these conserved quantities break as soon as one considers trajectories outside the regular islands.

1) You are right.

Generically Hamiltonian systems have local complete solutions:

In the general theory of partial differential equations of Hamilton–Jacobi type, a complete solution (i.e. one that depends on n independent constants of integration, where n is the dimension of the configuration space), exists in very general cases, but only in the local sense.

That's more detailedly explained in Qmechanic's answers to the questions Integrable vs. Non-Integrable systems and Constants of motion vs. integrals of motion vs. first integrals. For instance:

It follows from Caratheodory-Jacobi-Lie theorem that every finite-dimensional autonomous Hamiltonian system on a symplectic manifold $(M,\omega)$ is locally maximally superintegrable [i.e., completely integrable] in sufficiently small local neighborhoods around any point of $M$ (apart from critical points of the Hamiltonian).
The main point is that (global) integrability is rare, while local integrability is generic.

2) Is it strange?

Probably the demonstration of the theorem mentioned above explains why we can find these local constant of integration. But, without going through it at the moment, the fact that it excludes critical points suggests it might simply follow from the system smoothness - after all, the chaotic sea is replete with unstable periodic orbits.

• Why does the theorem exclude critical points of the Hamiltonian? Consider, for example, a local minimum of $H$. In the neighborhood of this point the Hamiltonian can be Taylor-expanded up to the second order thus obtaining an oscillator-like hamiltonian which is of course integrable. In other words: I expected that regular orbits could be found primarily around maxima and minima of $H$. Jul 13, 2018 at 8:48
• @AndreaPaco I think the exception is made for two reasons. First, critical points almost always have anyway to be handled with care in demonstrations - and note that even in the center-type critical point you mention, the movement at the critical point is qualitatively different from the oscillations around it. Second, critical points can also be saddles (unstable equilibria), which are arguably even less smooth. Jul 13, 2018 at 9:10
• Can you please exapand your first point? Jul 13, 2018 at 9:12
• I doesn't have much depth, really. It's simply that many theorems assume at some point smoothness or a nonzero derivative, so they aren't valid in general when these conditions aren't fulfilled. Jul 13, 2018 at 9:20