Two tales of a rigorous Derivation of the Hamiltonian Structure

Nataša Pavlovic, UT Austin
Fine Hall 314

In-Person Talk 

Many mathematical works have focused on understanding the manner in which the dynamics of a nonlinear  equation arises as an effective equation. 

  • For example, the cubic nonlinear Schrodinger equation (NLS) is an effective equation for a system of N bosons interacting pairwise via a delta or approximate delta potential. In this talk, we will advance a new perspective on deriving an effective equation, which focuses on structure. In particular, we will show how the Hamiltonian structure for the cubic NLS in any dimension arises from the corresponding structure at the N-particle level.
  • On the other hand, the Vlasov equation in any spatial dimension has long been known to be an infinite-dimensional Hamiltonian system whose bracket structure is of Lie-Poisson type. In parallel, it is classical that the Vlasov equation is a mean-field limit for a pairwise interacting Newtonian system. Motivated by this knowledge, we provide a rigorous derivation of the Hamiltonian structure of the Vlasov equation, both the Hamiltonian functional and Poisson bracket, directly from the many-body problem. This work settles a question of Marsden, Morrison, and Weinstein on providing a ``statistical basis'' for the bracket structure of the Vlasov equation.

The talk is based on joint works with Dana Mendelson, Joseph Miller, Andrea Nahmod, Matthew Rosenzweig and Gigliola Staffilani.