Related papers: Cylindrical type integrable classical systems in a…
A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New…
Motion of a charged particle in uniform magnetic field has been studied in detail, classically as well as quantum mechanically. However, classical dynamics of a charged particle in non-uniform magnetic field is solvable only for some…
A system of two particles with spin s=0 and s=1/2 respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order integrals of…
We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically…
Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an…
We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,\phi)=\omega^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k \phi}}+ \frac{\beta k^2}{r^2 \sin^2 {k \phi}}$ are closed for all integer and rational…
We construct complete sets of invariant quantities that are integrals of motion for two Hamiltonian systems obtained through a reduction procedure, thus proving that these systems are maximally superintegrable. We also discuss the reduction…
The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate its time evolution and the corresponding action-angle…
In this paper an approach is proposed to represent a class of dissipative mechanical systems by corresponding infinite-dimensional Hamiltonian systems. This approach is based upon the following structure: for any non-conservative classical…
Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the integrability of such…
We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which…
This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D…
In this paper we give examples of applications of general methods of quantization by symmetrization of classical integrable systems, which have been illustrated in two previous works by the same authors. We consider two classes of systems…
Classical and quantum superintegrable systems have a long history and they possess more integrals of motion than degrees of freedom. They have many attractive properties, wide applications in modern physics and connection to many domains in…
Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the…
It is noted that the Schrodinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order…
We introduce a new 2N--parametric family of maximally superintegrable systems in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator in a 2N--dimensional configuration space. These systems possess closed bounded…
We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate…
The quantum $H_3$ integrable system is a 3D system with rational potential related to the non-crystallographic root system $H_3$. It is shown that the gauge-rotated $H_3$ Hamiltonian as well as one of the integrals, when written in terms of…
We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth and sixth order in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets and…