I Does the Hamilton-Jacobi equation exist for chaotic systems?

[andresB]

TL;DR Summary

Does the Hamilton-Jacobi equation exist for chaotic systems?

Given a Hamiltonian $H(\mathbf{q},\mathbf{p})$, in the time-independent Hamilton-Jacobi approach we look for a canonical transformation $(\mathbf{q},\mathbf{p})\rightarrow(\mathbf{Q},\mathbf{P})$ such that the new Hamiltonian is one of the new momenta $$H(\mathbf{q},\mathbf{p})=K(\mathbf{Q},\mathbf{P})=P_{1}=E.$$
If such transformation exists, all the momenta $\mathbf{P}$ are constant of the motion. And, since the transformation is canonical, we will have n constant of the motion in involution, i.e., $\left\{ P_{i},P_{j}\right\} =0.$ But this seems to be the requirement of the Liouville theorem for integrability. Chaotic systems don't have that many constants of motion in involution.
This seems to imply that the Hamilton-Jacobi equations cannot be even written for chaotic systems, so my reasoning has to be wrong somewhere. Where?

 

[wrobel]

TL;DR Summary: Does the Hamilton-Jacobi equation exist for chaotic systems?

Given a Hamiltonian H(q,p), in the time-independent Hamilton-Jacobi approach we look for a canonical transformation (q,p)→(Q,P) such that the new Hamiltonian is one of the new momenta H(q,p)=K(Q,P)=P1=E.
If such transformation exists, all the momenta P are constant o

Such a transformation exists locally outside an equilibrium. The set where this transformation is defined is not invariant. The system comes and goes away from there, while a dynamical chaos appears in invariant sets.

 

[andresB]

Such a transformation exists locally outside an equilibrium. The set where this transformation is defined is not invariant. The system comes and goes away from there, while a dynamical chaos appears in invariant sets.

Ok, I suspected the issue had to do with the new momenta not being defined globally.

Yet, I'm not sure I understand your answer. Can you give more details or point to a source with an example?

 

[wrobel]

A dynamical chaos is an informal concept which expresses a complex of very different effects. The common feature for all these effects is a complicated behavior of trajectories of a dynamical system. Every time, one should specify what he means by saying “dynamical chaos”. For example, in Hamiltonian systems, ergodicity or separatrix splitting are commonly considered as a dynamical chaos, but there are a lot of other chaotic effects. Real life examples are really hard.
See for example this https://link.springer.com/book/10.1007/978-3-642-03028-4
or start from this https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf

UPD

Dissipative systems commonly have attractors. If an attractor is of complicated geometry, say has a fractional dimension, then trajectories which wind up on the attractor replicate its geometry and become complicated. That is another chaotic effect.

 

[andresB]

Fascinating and frustrating at the same time.

See for example this https://link.springer.com/book/10.1007/978-3-642-03028-4

My doubts started by studyng perturbation theory. In the book Canonical Perturbation Theories Degenerate Systems and Resonance, I find the following

The usual books in analytical mechanics don't deal with this.