Determination of Lagrangians from equations of motion and commutator brackets

The question as to whether a given set of equations, which govern the dynam ical evolution of a system, determine a Lagrangian is considered. This prob lem, which has come to be known as the inverse problem of the calculus of v ariations, is reviewed and theorems which contain systems of partial differ ential equations which determine a type of self-adjointness are developed. It is shown that, given a reasonable form for the classical correspondence, the usual quantum commutator brackets can be expressed in terms of classic al quantities which satisfy a particular form of these equations.