Symmetries and invariances in classical physics

Abstract

The group theoretical notion of symmetry is the notion of invariance under a specified group of transformations. “Invariance” is a mathematical term: something is invariant when it is left unaltered by a given transformation. This mathematical notion is used to express the notion of physical symmetry. The chapter discusses the distinction between symmetries of objects and of laws, and that between symmetry principles and symmetry arguments. It includes a discussion of Curie's principle. The important connection between symmetries, as studied in physics and the mathematical techniques of group theory is discussed. A brief history is given on how group theory was applied first to geometry and then to physics in the course of the nineteenth century, preluding to the central importance acquired by group theoretical techniques in contemporary physics. The chapter focuses on the roles and meaning of symmetries in these theories, which leads into the discussion of Noether's theorems.