Sistemas hamiltonianos integráveis em órbitas coadjuntas de grupos de Lie clássicos

Abstract

In this work, we will study Hamiltonian systems on (co)adjoint orbits of Lie groups. In particular, our focus lies in examining issues concerning integrability, as it allows for exact analytical solutions to the equations of motion. We will only consider Hamiltonian functions of a specific type, namely, defined through a momentum map. One of the objectives is to investigate a concrete construction of integrable Hamiltonian systems on (co)adjoint orbits using Lax formalism. The methodology employed is based on the application of the Thimm trick. To this end, we should see (co)adjoint orbits of compact Lie groups as symplectic manifolds and determine a momentum map associated with the restriction of the (co)adjoint action to these orbits. The equivariance condition of the momentum map will be essential for constructing involutive functions for the system.