Hamiltonian Motions Under Strong Constrains

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

The project focuses on Hamiltonian ODEs and PDEs under strong constraining forces. For an given Hamiltonian composed of the kinetic energy and a potential energy, consider particles restricted to a submanifold (independent of time) in the configuration space, i.e. a holonomic constraint. The idealization of the constrained particles' motion is governed by a Hamiltonian system defined on the tangent bundle of this submanifold involving geometric notions. In physics, an alternative way to realize the constraint is to consider the system in the original space with an extra strong potential which penalizes the distance to the constraining submanifold. This idea applies to both Hamiltonian ODEs and PDEs. While the convergence of the motions under strong constraining potentials to the limit geometric Hamiltonian motions on finite time intervals has been studied for ODEs, the problem in PDEs is basically untouched. The project focuses on two questions. The first is the convergence of the strongly penalized motions, both on finite and infinite time intervals. The second question is the relation between the dynamics of the strongly penalized motions and their limits, i.e. the structural stability under strong constraining forces. The subjects are stability, periodic motions, homoclinic motions, resonances, etc. The problem can also be viewed as homogenization or elliptic type singular perturbations. The problem of motions of particles restricted to submanifolds in the configuration space appears naturally in both classical mechanics and PDEs. For example, in classical mechanics, whenever a rigid rod is considered as elastic with a large elastic coefficient, the problem falls in this category. Also, it is found, in material science, that some anti-ferromagnetic systems converges to geometric wave equations targeted on the unit 2-dimensional sphere formally. Therefore, it is important to study how the constrained motions converge. Moreover, as the penalized motions have high frequency oscillations, it is even more important to investigate the relation between the asymptotic qualitative behaviors.