Construction of an Invariance from a Conservation Law, and Vice Versa, in Classical Mechanics

Abstract

Special classes of problems (each class is called a mechanics) are discussed for which it is either always possible to construct a conservation law when one is given an invariance of the equations of motion, or else it is always possible to construct an invariance when one is given a conservation law. Of chief interest is the introduction of a set of mechanics obeying the latter alternative in the form: (A) Given any constant of the motion, one can always construct a linear combination of its first derivatives (with fixed constant coefficients) which will be an invariance generator. Hamiltonian mechanics is, roughly speaking, ``derived'' as a consequence of (A) and the additional postulate: (B) One can always find a function from which, when one constructs the linear combination of its first derivatives, one obtains the motion generator. Another interesting mechanics satisfying (A) [but not (B)] is discussed. This is a class of harmonic-oscillator problems, called oscillator mechanics. It is shown that there is a class of problems belonging to both Hamiltonian mechanics and oscillator mechanics. This class of problems has an algebra that is the same as the algebra of quantum mechanics.