Action in Classical Mechanics

Jack Cohn

The problem of the reduction of quantum mechanics to classical mechanics is considered. It is demonstrated that the customary derivation of classical mechanics from the Schrödinger equation via the Hamilton-Jacobi action equation is incomplete. An improved derivation, involving the expectation value of the action, rather than the action itself, is given. Under this formulation the Hamilton-Jacobi description of classical mechanics is obtained from quantum mechanics, for suitable wave packets as h goes to zero.

G.C. Stey

Newton C. A. da Costa, Francisco Antônio Dória

Richard P. Feynman

A generalization of quantum mechanics is given in which the central mathematical concept is the analogue of the action in classical mechanics. It is therefore applicable to mechanical systems whose equations of motion cannot be put into Hamiltonian form. It is only required that some form of least action principle be available. It is shown that if the action is the time integral of a function of velocity and position (that is, if a Lagrangian exists), the generalization reduces to the usual form of quantum mechanics. In the classical limit, the quantum equations go over into the corresponding classical ones, with the same action function. As a special problem, because of its application to electrodynamics, and because the results serve as a confirmation of the proposed generalization, the interaction of two systems through the agency of an intermediate harmonic oscillator is discussed in detail. It is shown that in quantum mechanics, just as in classical mechanics, under certain circumstances the oscillator can be completely eliminated, its place being taken by a direct, but, in general, not instantaneous, interaction between the two systems. The work is non-relativistic throughout.

R. Shankar

L. M. Garrido

In this paper we consider a general action principle for mechanics written by means of the elements of a Lie algebra. We study the physical reasons why we have to choose precisely a Lie algebra to write the action principle. By means of such an action principle we work out the equations of motion and a technique to evaluate perturbations in a general mechanics that is equivalent to a general interaction picture. Classical or quantum mechanics come out as particular cases when we make realizations of the Lie algebra by derivations into the algebra of products of functions or operators, respectively. Later on we develop in particular the applications of the action principle to classical and quantum mechanics, seeing that in this last case it agrees with Schwinger's action principle. The main contribution of this paper is to introduce a perturbation theory and an interaction picture of classical mechanics on the same footing as in quantum mechanics.