New geometrical frameworks for classical impulsive mechanics

A new context is introduced to give a formal geometric environment for the study of impulsive mechanics of systems with finite number of degrees of freedom. The new structures embody the usual ones of analytical mechanics such as tangent bundles for time independent systems and jet-bundles for time dependent ones. They allow a causally consistent definition of active impulse acting on the systems and a clear geometric view of the equations of motion.     

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.     

Nilpotent classical mechanics:s-geometry

We introduce specific type of hyperbolic spaces. It is not a general linear covariant object, but is of use in constructing nilpotent systems. In the present work necessary definitions and relevant properties of configuration and phase spaces are indicated. As a working example we use aD=2 isotropic harmonic oscillator.     

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.     

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.     

Quantum mechanics as a classical diffusion process

A direct construction is provided showing that the classical expectation value for the energy of a particle executing a classical diffusion process is equivalent to the quantum mechanical form with a Hamiltonian structured like that for a charged particle in an electromagnetic field.     

Nonlinear quantum mechanics is a classical theory

Quantum mechanics with nonlinear operators is shown to be an essentially classical theory. A general scheme of delinearization of a quantum theory is described.     

The transfer matrix: A geometrical perspective

We present a comprehensive and self-contained discussion of the use of the transfer matrix to study propagation in one-dimensional lossless systems, including a variety of examples, such as superlattices, photonic crystals, and optical resonators. In all these cases, the transfer matrix has the same algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as well as the group of unimodular real matrices underlying the structure of the abcd$abcd$ law, which explains many subtle details. We elaborate on the geometrical interpretation of the transfer-matrix action as a mapping on the unit disk and apply a simple trace criterion to classify the systems into three types with very different geometrical and physical properties. This approach is applied to some practical examples and, in particular, an alternative framework to deal with periodic (and quasiperiodic) systems is proposed.     

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.     

对一个力学碰撞问题的讨论

对普通物理力学中有关碰撞的一个问题,从恢复系数e的取值角度出发,进行了详细求解和讨论,得到了更完整的结果.     

Quantum mechanics in a discrete model of classical physics

The role of probability theory in classical physics is examined. It is found that the probabilities for the outcomes of typical experiments depend strongly on the assumed behavior of given classical models at infinity. A discrete classical model is introduced and it is shown that the resulting probabilities are similar to those in the usual theory of quantum mechanics.     

An underlying geometrical manifold for Hamiltonian mechanics

We show that there exists an underlying manifold with a conformal metric and compatible connection form, and a metric type Hamiltonian (which we call the geometrical picture), that can be put into correspondence with the usual Hamilton–Lagrange mechanics. The requirement of dynamical equivalence of the two types of Hamiltonians, that the momenta generated by the two pictures be equal for all times, is sufficient to determine an expansion of the conformal factor, defined on the geometrical coordinate representation, in its domain of analyticity with coefficients to all orders determined by functions of the potential of the Hamiltonian–Lagrange picture, defined on the Hamilton–Lagrange coordinate representation, and its derivatives. Conversely, if the conformal function is known, the potential of a Hamilton–Lagrange picture can be determined in a similar way. We show that arbitrary local variations of the orbits in the Hamilton–Lagrange picture can be generated by variations along geodesics in the geometrical picture and establish a correspondence which provides a basis for understanding how the instability in the geometrical picture is manifested in the instability of the the original Hamiltonian motion.     

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.