The Variational Principle and natural transformations I. Autonomous dynamical systems

Abstract

When a system of differential equations is derivable from a variational principle, an outstanding consequence is the connection which ensues between symmetries and conservation laws. The idea behind the present work is to use this consequence as a new characterization of the variational principle itself. For general systems of equations there is a rather weak form of the connection, as will be demonstrated; but it has highly arbitrary and non-invariant features, and, for instance, there is not a formula associating a particular infinitesimal invariance with a particular constant of the motion. In terms of the modern technical concept which will be employed, the transformation is not natural. However, if the system is derivable from a variational principle, there follows a preferred transformation; and it has certain invariance properties which accord with the concept of naturality.