Abstract

In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.