The Classification of Finite-Dimensional Lie Algebras of Hamiltonian Vector Fields on the Plane

Explore the classification of finite-dimensional Lie algebras of Hamiltonian vector fields on the plane in this comprehensive mathematical physics seminar. Delve into Sophus Lie's classification of locally diffeomorphic finite-dimensional Lie algebras of vector fields on the plane, following the modern approach by Gonzalez-Lopez, Kamran, and Olver. Examine the relationships between different Lie algebras in the classification, including which are diffeomorphic to Lie subalgebras of others. Investigate the subclass of Lie algebras that are Hamiltonian relative to a symplectic structure. Learn how to apply this classification to study relevant types of Hamiltonian systems on the plane and related results. Gain insights into the mathematical foundations of Lie-Hamilton systems and their applications in physics and mathematics.