VARIATIONAL CALCULUS, SYMMETRIES AND REDUCTION

Abstract

Variational Principles and Symmetries are both fundamental ingredients of the most relevant (either classical or modern) physical theories, and of many interesting models in Pure and Applied Mathematics. While the variational principle determines the law (differential and sometimes algebraic equations) of the dynamics, the symmetries give way to reduction. Noether's theorem connects some symmetries with constants of the motion in the framework of Lagrangian (point symmetries) or Hamiltonian (canonical symmetries) systems, becoming a powerful tool in the reduction process. This entanglement between variational principles and symmetries is the source of a very rich machinery that holds outstanding mathematical beauty and physical interpretations; however, it sometimes generates misunderstandings about the different roles played by each of these geometric ingredients in the framework of the theory. The aim of this paper is to review some basic facts about the geometric anatomy and physiology of classical dynamical systems ruled by a variational principle and subject to symmetries, distilling the inherited properties of the system according to its cause. A brief discussion of some illustrative examples accompany the exposition to enhance the main ideas.