Symmetry Groups and Conservation Laws

Abstract

In the study of systems of differential equations, the concept of a conservation law, which is a mathematical formulation of the familiar physical laws of conservation of energy, conservation of momentum and so on, plays an important role in the analysis of basic properties of the solutions. In 1918, Emmy Noether proved the remarkable result that for systems arising from a variational principle, every conservation law of the system comes from a corresponding symmetry property.† For example invariance of a variational principle under a group of time translations implies the conservation of energy for the solutions of the associated Euler-Lagrange equations, and invariance under a group of spatial translations implies conservation of momentum. This basic principle constitutes the first fundamental result in the study of classical or quantum-mechanical systems with prescribed groups of symmetries. Moreover, Noether’s method is the only really systematic procedure for constructing conservation laws for complicated systems of partial differential equations.KeywordsSymmetry GroupVariational ProblemInfinitesimal GeneratorVariational SymmetryEuler Operator