New integrable problem of classical mechanics

Complete integrability in Liouville's sense is proven for rotation of an arbitrary rigid body with a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. A consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects (in the second approximation). Explicit formulae are obtained expressing angular velocities of the rigid body in terms of -functions for Riemannian surfaces. Integrable cases are found for rotation of a rigid body in nonlinear Newtonian potential fields.     

Two-dimensional supersymmetry: From SUSY quantum mechanics to integrable classical models

Two known two-dimensional SUSY quantum mechanical constructions—the direct generalization of SUSY with first-order supercharges and higher-order SUSY with second-order supercharges—are combined for a class of 2-dim quantum models, which are not amenable to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained—the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton-Jacobi method for related “flipped” potentials are established.     

Generalized classical mechanics

Generalized classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. The Lagrangian of generalized classical mechanics has been introduced, and equation of motion has been obtained. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented. Oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of generalized classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of generalized classical mechanics and the well-known fundamental equations of classical mechanics.     

Supersymmetric classical mechanics

The purpose of the paper is to construct a supersymmetric Lagrangian within the framework of classical mechanics which would be regarded as a candidate for passage to supersymmetric quantum mechanics. The authors felicitate Prof. D S Kothari on his eightieth birthday and dedicate this paper to him on this occasion.     

Quantum mechanics as an integrable system

Basic dynamical equations of quantum mechanics, including the Schrödinger, Pauli, Dirac, and Klein-Gordon equations, are bi-hamiltonian systems with an infinite number of conservation laws.     

N-dimensional classical integrable systems from Hopf algebras

A systematic method to constructN-body integrable systems is introduced by means of phase space realizations of universal enveloping Hopf algebras. A particular realization for theso(2, 1) case (Gaudin system) is analysed and an integrable quantum deformation is constructed by using quantum algebras as Poisson-Hopf symmetries.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

Quantum mechanics as a deformation of classical mechanics

Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.Work supported in part by the National Science Foundation.     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.     

Statistical mechanics of classical systems

An algebra of thermodynamic operators of the fluctuations of physical quantities in classical statistical mechanics is found and its properties studied. A method is proposed for obtaining equations that describe the equilibrium and nonequilibrium statistical ensembles of classical systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 6–11, May, 1980.     

Probabilistic formulation of classical mechanics

Starting axiomatically with a system of finite degrees of freedom whose logic _c is an atomic Boolean -algebra, we prove the existence of phase space _c, as a separable metric space, and a natural (weak) topology on the set of statesI (all the probability measures on _c) such that _c, the subspace of pure statesP, the set of atoms of _c and the spaceP( _c) of all the atomic measures on _c, are all homeomorphic. The only physically accessible states are the points of _c. This probabilistic formulation is shown to be reducible to a purely deterministic theory.     

On the classical limit of Berry's phase integrable systems

Berry's Phase is given by integration of a characteristic two form. We consider integrable systems defined by Weyl quantized classical Hamiltonians. It is shown that the limit of /i times this tow form is the curvature of the classical connection whose holonomy is the Hannay angels. A result of this type was derived by Berry [B2].     

Quasigraded deformations of loop algebras and classical integrable systems

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras. Using them we obtain new series of integrable Hamiltonian systems on semisimple Lie algebras and their extensions. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Discrete effects in classical mechanics

By utilizing some results of discrete mechanics, effects related to the discretization of the equations of motion of classical mechanics are analyzed. Some examples of periodic motions are developed and it is shown that the discrete frequencies are different from the continuous one. Numerical calculations involving orbits are presented. A numerical algorithm, suggested by discrete mechanics, is compared with conventional methods of second order.     

BRST cohomology in classical mechanics

The classical (non-quantum) cohomology of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry in phase space is defined and worked out. No group action for the gauge transformations is assumed. The results cover, therefore, the general case of an open algebra and are valid off-shell. Each cohomology class contains all BRST invariant functions with fixed ghost number (an integer) which differ from each other by a BRST variation. If the ghost number is negative there is only the trivial class whose elements are equivalent to zero. If the ghost number is positive or zero there is a bijective correspondence between the BRST classes and those of the exterior derivative along the gauge orbits. These gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish. The BRST invariant functions of ghost numberp are then related to closedp-forms along the orbits. The addition of a BRST variation corresponds to the addition of an exact form. Some comments about the quantum case are included. The physical meaning of the classes with ghost number greater than zero is not discussed.Chercheur qualifié du Fonds National de la Recherche Scientifique (Belgium)     

The S-matrix in classical mechanics

In the Hilbert-space version of classical mechanics, scattering theory forN-particle systems is developed in close analogy to the quantum case. Asymptotic completeness is proved for forces of finite range. Infinite-range forces lead to the problem of stability of bound states and can be dealt with only in some simple cases.It is a pleasure to thank Prof.L. Motchane for his kind hospitality at the I.H.E.S., where most of this work was done, and where the author profited from discussions withD. Ruelle andO. E. Lanford.