The Feynman problem and the inverse problem for Poisson dynamics

Abstract

We review the Feynman proof of the Lorentz force equations, as well as its generalization to the dynamics of particles with internal degrees of freedom. In addition, we discuss the inverse problem for Poisson dynamics and the inverse problem of the calculus of variations. It is proved that the only classical dynamics compatible with localizability and the existence of second order differential equations on tangent bundles over arbitrary configuration spaces is necessarily of the Lagrangian type. Furthermore, if the dynamics is independent of the velocity of test particles, it must correspond to that of a particle coupled to an electromagnetic field and/or a gravitational field. The same ideas are carried out for particles with internal degrees of freedom. In this case, if we insist on a weak localizability condition and the existence of a second order Hamiltonian differential equation, then the dynamics results from a singular Lagrangian. (Here we assume in addition that the dynamics satisfies a regularity condition.) These results extend those of Feynman and provide the conditions which guarantee the existence of a Lagrangian description. They are applied to systematically discuss Feynman's problem for systems possessing Lie groups as configuration spaces, with internal variables modeled on Lie algebras of groups. Finally, we illustrate what happens when the condition of localizability is dropped. In this regard, we obtain alternative Hamiltonian descriptions of standard dynamical systems. These non-standard solutions are discussed within the framework of Lie-Poisson structures.