Ian Anderson (Utah State University, USA): Symmetries, Conservation Laws and Variational Principles
Channel: Centrum Fizyki Teoretycznej PAN
Date: 23/01/2023

Prof. Ian Anderson (Utah State University, USA).
Symmetries, Conservation Laws and Variational Principles
Abstract: In this series of lectures I will discuss the mathematics around the celebrated Noether Theorems in the calculus of variations.

A. Symmetries
1. External Symmetries
2. Internal Symmetries
3. Generalized Symmetries
4. Examples
5. Symbolics

B. Conservation Laws
1. First Integrals of ODE
2. Classical Conservation Laws
3. Lower Degree Conservation Laws
4. Examples
5. Intermediate Integrals
6. Characteristic Lagrangians
7. Conservation Laws in General Relativity
8. Characteristic Cohomology
9. Integral Invariants

C. Noether Theorems
1. Variational Calculus
2. Noether I
3. Natural Operators
4. Noether II
5. Examples
6. Noether III
7. Noether IV

D. The Taken’s Problem (An Inverse Noether Theorem)
1. Statement
2. A Simple Case
3. The Gelfand-Dickii Transform
4. A Counterexample
5. Further Results
6. Divergence-free Tensors

E. Invariant Variational Problems and Differential Invariants

This is a bi-weekly seminar for the Grieg collaboration project SCREAM: Symmetry, Curvature Reduction, and EquivAlence Methods, funded by the Norwegian Financial Mechanism 2014-2021. The registration number of the project is 2019/34/H/ST1/00636. 

The focus of the project is on a broad class of differential geometric structures known as Cartan and parabolic geometries, which include conformal, projective, CR, and ODE geometry, (2, 3, 5)-distributions, parabolic contact structures, and many more besides. Moreover, we explore the interaction of these structures with a variety of subjects such as mechanical systems and geometric robots, the theory of integrable systems, and Penrose's Conformal Cyclic Cosmology programme. 

More information can be found in www.cft.edu.pl/grieg.