In mathematics, a superintegrable Hamiltonian system is a
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
on a
-dimensional
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
for which the following conditions hold:
(i) There exist
independent integrals
of motion. Their level surfaces (invariant submanifolds) form a fibered manifold
over a connected open subset
.
(ii) There exist smooth real functions
on
such that the
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
of integrals of motion reads
.
(iii) The matrix function
is of constant corank
on
.
If
, this is the case of a
completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on
action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion witho ...
of completely integrable Hamiltonian system as follows.
Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold
is a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
in tori
. There exists an open neighbourhood
of
which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates
,
,
such that
are coordinates on
. These coordinates are the
Darboux coordinates
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
on a symplectic manifold
. A Hamiltonian of a superintegrable system depends only on the action variables
which are the Casimir functions of the coinduced
Poisson structure
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule
: \ = \h + g \ .
Equivalen ...
on
.
The
Liouville-Arnold theorem for
completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder
.
See also
*
Integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
*
Action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion witho ...
*
Nambu mechanics
In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are sym ...
*
Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For ...
References
* Mishchenko, A., Fomenko, A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978) 113.
* Bolsinov, A., Jovanovic, B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003) 305; .
* Fasso, F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87(2005) 93.
* Fiorani, E.,
Sardanashvily, G., Global action-angle coordinates for completely integrable systems with non-compact invariant manifolds, J. Math. Phys. 48 (2007) 032901; .
* Miller, W., Jr, Post, S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001,
* Giachetta, G., Mangiarotti, L.,
Sardanashvily, G., ''Geometric Methods in Classical and Quantum Mechanics'' (World Scientific, Singapore, 2010) ; {{arXiv, 1303.5363.
Hamiltonian mechanics
Dynamical systems
Integrable systems