Superintegrable Hamiltonian system

Last updated August 07, 2019

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a $2n$ -dimensional symplectic manifold for which the following conditions hold:

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 be studied in both Hamiltonian mechanics and dynamical systems theory.

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

(i) There exist $k>n$ independent integrals $F_{i}$ of motion. Their level surfaces (invariant submanifolds) form a fibered manifold $F:Z\to N=F(Z)$ over a connected open subset $N\subset \mathbb {R} ^{k}$ .

(ii) There exist smooth real functions $s_{ij}$ on $N$ such that the Poisson bracket of integrals of motion reads $\{F_{i},F_{j}\}=s_{ij}\circ F$ .

In geometry, a Poisson structure on a smooth manifold $is a Lie bracket on the algebra of smooth functions on, subject to the Leibniz rule$

(iii) The matrix function $s_{ij}$ is of constant corank $m=2n-k$ on $N$ .

If $k=n$ , 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 of completely integrable Hamiltonian system as follows.

In the context of differential equations to integrate an equation means to solve it from initial conditions. Accordingly, an integrable system is a system of differential equations whose behavior is determined by initial conditions and which can be integrated from those initial conditions.

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 frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus.

Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold $F$ is a fiber bundle in tori $T^{m}$ . There exists an open neighbourhood $U$ of $F$ which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates $(I_{A},p_{i},q^{i},\phi ^{A})$ , $A=1,\ldots ,m$ , $i=1,\ldots ,n-m$ such that $(\phi ^{A})$ are coordinates on $T^{m}$ . These coordinates are the Darboux coordinates on a symplectic manifold $U$ . A Hamiltonian of a superintegrable system depends only on the action variables $I_{A}$ which are the Casimir functions of the coinduced Poisson structure on $F(U)$ .

Fiber bundle continuous surjection satisfying a local triviality condition

In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space E and a product space $is defined using a continuous surjective map$

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 them being symplectic geometry. The theorem is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem.

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 $T^{m-r}\times \mathbb {R} ^{r}$ .

Related Research Articles

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

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. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself $as one of the new canonical momentum coordinates.$

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.

In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle $of a manifold . In physics, it is used to create a correspondence between the velocity of a point in a mechanical system to its momentum, thus providing a bridge between Lagrangian mechanics with Hamiltonian mechanics.$

Dynamical billiards Dynamical system abstract an ideal game of billiards, with elastic collisions off boundaries

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. Dynamical billiards may also be studied on non-Euclidean geometries; indeed, the very first studies of billiards established their ergodic motion on surfaces of constant negative curvature. The study of billiards which are kept out of a region, rather than being kept in a region, is known as outer billiard theory.

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Hamiltonian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion

In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.

Gennadi Sardanashvily was a theoretical physicist, a principal research scientist of Moscow State University.

In mathematics, a Lagrangian system is a pair $(Y, L)$ , consisting of a smooth fiber bundle $Y \to X$ and a Lagrangian density $L$ , which yields the Euler–Lagrange differential operator acting on sections of $Y \to X$ .

References

Mishchenko, A., Fomenko, A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978) 113. doi : 10.1007/BF01076254
Bolsinov, A., Jovanovic, B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003) 305; arXiv : math-ph/0109031.
Fasso, F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87(2005) 93. doi : 10.1007/s10440-005-1139-8
Fiorani, E., Sardanashvily, G., Global action-angle coordinates for completely integrable systems with non-compact invariant manifolds, J. Math. Phys. 48 (2007) 032901; arXiv : math/0610790.
Miller, W., Jr, Post, S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001, doi : 10.1088/1751-8113/46/42/423001 arXiv : 1309.2694
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Methods in Classical and Quantum Mechanics (World Scientific, Singapore, 2010) ISBN 978-981-4313-72-8; arXiv : 1303.5363.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

Superintegrable Hamiltonian system

See also

Related Research Articles

References