Integrable Hamiltonian systems: generalization to a case of non-compact invariant submanifolds (from my Scientific Biography)

The study of integrable Hamiltonian systems in conservative mechanics did not lay in the mainstream of my research, and they were in my field of vision by accident. Moreover, it was difficult to imagine a possibility of generalization of the fundamental Liouville - Arnold theorem on "action-angle" coordinates in a neighborhood of an invariant submanifolds of a completely integrable Hamiltonian system.

This theorem was proved for a case of compact invariant submanifolds. First, it is proved that a compact invariant submanifold is a multi-dimensional torus, and then, this fact is used in a simple way  that every function on a torus is cyclic.

It seemed to me that this condition can be avoided. Not assuming initially a compactness of an invariant submanifold of an integrable Hamiltonian system, we proved that it is a multi-dimensional cylinder, and then managed to build generalized  "action-angle" coordinates in its neighborhood [102,103]. It all took less than a month. Bring the chronology of events related to that.

It was further naturally to go to partially integrable Hamiltonian systems and, in 2003, we generalized the Nekhoroshev theorem to the case of noncompact  invariant submanifolds [106,108]. In connection with them, we considered bi-Hamiltonian systems, and described a class of Poisson structures with respect to which a Hamiltonian system is partially integrable [108].

As integrable Hamiltonian systems are still, not my subject, at that time I did not suspect about existence of superintegrable Hamiltonian systems. They caught me in the eyes in 2006, and we have generalized the Mishchenko - Fomenko theorem to the case of non-compact invariant submanifolds [123]. We used the fact that such a submanifold in fact is an invariant submanifold of a partially integrable Hamiltonian system, and referred  to our generalization of the Nekhoroshev theorem.

Our results touched generalized "action-angle" coordinates in some neighborhood of non-compact invariant submanifolds. There were known topological obstructions to the existence of global "action-angle" coordinates for completely integrable Hamiltonian systems with compact invariant submanifolds. We generalized these results to a non-compact case [125]. Moreover, it turned out that, in a general case, a phase space of a superintegrable system system is decomposed into open areas, where a system is different, i.e., its integrals of motion form different Lie algebra [135]. An example is the Kepler system whose phase space is split into two areas. In one of them, invariant submanifolds are ellipses, and integrals of motion form the Lie algebra so(3), but in the other, they are hyperboles, and the Lie algebra of integrals of motion is so(2,1).

An example of integrable Hamiltonian systems with non-compact invariant submanifolds are non-autonomous integrable Hamiltonian systems whose invariant submanifolds obviously contain the time axis R. The theory of such integrable Hamiltonian systems has been developed [17,103].

Using the method of geometrical quantization, we have implemented quantization of completely integrable and superintegrable Hamiltonian systems in "action-angle" variables [104,124], including non-autonomous completely integrable systems [102]. It should be noted that, since transformations between original variables and "action-angle" variables are non-linear, quantization in those and other variables are not equivalent. However, as already noted, in "action-angle" variables, we can build non-adiabatic classical and quantum holonomy operators for completely integrable Hamiltonian system [15,17,112].

Reference:

G.Sardanashvily, My Scientific Biography