Symmetries, Conservation Laws and Variational Principles

Explore the mathematics behind the renowned Noether Theorems in the calculus of variations through this comprehensive lecture by Prof. Ian Anderson from Utah State University. Delve into the intricate world of symmetries, conservation laws, and variational principles across five main topics. Begin with an in-depth examination of external, internal, and generalized symmetries, complete with examples and symbolic representations. Progress to conservation laws, covering first integrals of ODE, classical and lower degree conservation laws, intermediate integrals, characteristic Lagrangians, and their applications in general relativity. Unpack the Noether Theorems, exploring variational calculus, natural operators, and multiple iterations of Noether's work. Investigate the Taken's Problem, an inverse Noether Theorem, including its statement, simple cases, and the Gelfand-Dickii Transform. Conclude with a study of invariant variational problems and differential invariants. This 54-minute lecture is part of the SCREAM project, focusing on Cartan and parabolic geometries and their interactions with mechanical systems, integrable systems, and cosmology.