Canonical Transformations and Commutators in the Lagrangian Formalism

Abstract

In this paper we clarify a number of questions connected with the Lagrangian formulation of canonical transformations and commutator brackets. We make a distinction between "regular" and "singular" theories, the latter having such a structure that the Euler equations cannot be solved uniquely with respect to the accelerations. For "regular" theories we show that the introduction of the Poisson bracket by Peierls, which is based on a variation of the Lagrangian, and the infinitesimal canonical transformations introduced by Bergmann and Schiller lead to equivalent results. For "singular" theories we show first that constants of the motion do not necessarily generate invariant transformations and that, generally speaking, the relationship between transformations and generators is not unique in either direction. Then we show that by restricting ourselves to invariant transformations and their generators we can define commutators between constants of the motion unambiguously. The resulting bracket expressions vanish whenever at least one of the commuted constants of the motion vanishes (is a secondary constraint). It turns out that these commutator brackets in the Lagrangian formalism are equivalent not to Poisson brackets but to (generalized) Dirac brackets. A possible quantization procedure is sketched in the concluding section.