Christopher Pollack | Applied Mathematics, University of Waterloo
Extending Noether’s Theorem to Symmetries of Euler-Lagrange Equations of Motion Using the Covariant Phase Space Method: Applications to Poisson-Lie Groups and Integrable Systems
The study of symmetries has forever played an invaluable role in the development physical theories. The dynamics admitted by modern physical theories with Lagrangian descriptions, such as general relativity and quantum field theory, are almost entirely restricted by the Lie group defining their symmetry. The connection between physics and symmetries is made even tighter by Noether’s first theorem which relates the existence of a symmetry of a Lagrangian with the existence of a dynamically conserved current and charge. The definition of a symmetry in this sense strictly means the invariance of the corresponding action functional under the governing Lie group transformation. We show that one can relax this invariance of the action functional and replace it with the weaker condition of invariance of the Euler-Lagrange equations of motion while still being able to construct a meaningful dynamically conserved current and scalar charge in a generalized Noether sense. Using the framework of the covariant phase space method we extend the usual Noether analysis to these Euler-Lagrange symmetries and show that the resulting current satisfies a generalized Hamiltonian flow equation, captures in what sense these transformations are symplectomorphisms of the dynamical systems phase space, and under certain conditions how the current may give rise to a conserved group valued momentum map. It is shown that if the current admits such a momentum map then the acting Lie group may be naturally promoted to a Poisson-Lie group, and that several other important results from Poisson-Lie group theory may be immediately translated over. An analysis of the Euler-Lagrange symmetries and Poisson-Lie structure admitted by a dual pair of non-linear sigma models describing the dynamics of an open string is then done.