Hamiltonian Normal Forms, Lie Series Method for Transformations - Infinite-Dimensional Hamiltonians

Explore Hamiltonian normal forms and the Lie series method for canonical transformations in this advanced dynamics lecture. Delve into the simplification of Hamiltonian functions using the Lie series approach, and examine a one-degree-of-freedom system as an illustrative example. Transition to the study of infinite-dimensional Hamiltonian systems, focusing on partial differential equations such as the shallow water equations and the Korteweg-de Vries (KdV) equation. Gain insights into advanced topics in nonlinear dynamics and Hamiltonian mechanics, including applications to fluid dynamics and wave propagation.

Normal forms for general ODEs.
Hamiltonian normal forms.
Lie transformation method for finding a canonical transformation.
Simplifying the Hamiltonian function via the Lie series method.
Example 1 degree of freedom system to simplify via normal forms.
Infinite-dimensional Hamiltonian systems (PDEs).
Example: shallow water equations (Korteweg-de Vries or KdV equation).