Hamiltonian Motions Under Strong Constrains

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The project focuses on Hamiltonian ODEs and PDEs under strong constrainingforces. For an given Hamiltonian composed of the kinetic energy and apotential energy, consider particles restricted to a submanifold(independent of time) in the configuration space, i.e. a holonomicconstraint. The idealization of the constrained particles' motion isgoverned by a Hamiltonian system defined on the tangent bundle of thissubmanifold involving geometric notions. In physics, an alternative way torealize the constraint is to consider the system in the original spacewith an extra strong potential which penalizes the distance to theconstraining submanifold. This idea applies to both Hamiltonian ODEs andPDEs. While the convergence of the motions under strong constrainingpotentials to the limit geometric Hamiltonian motions on finite timeintervals has been studied for ODEs, the problem in PDEs is basicallyuntouched. The project focuses on two questions. The first is theconvergence of the strongly penalized motions, both on finite and infinitetime intervals. The second question is the relation between the dynamics ofthe strongly penalized motions and their limits, i.e. the structuralstability under strong constraining forces. The subjects are stability,periodic motions, homoclinic motions, resonances, etc. The problem can alsobe viewed as homogenization or elliptic type singular perturbations.The problem of motions of particles restricted to submanifolds in theconfiguration space appears naturally in both classical mechanics andPDEs. For example, in classical mechanics, whenever a rigid rod isconsidered as elastic with a large elastic coefficient, the problem fallsin this category. Also, it is found, in material science, that someanti-ferromagnetic systems converges to geometric wave equationstargeted on the unit 2-dimensional sphere formally. Therefore, it isimportant to study how the constrained motions converge. Moreover, as thepenalized motions have high frequency oscillations, it is even moreimportant to investigate the relation between the asymptotic qualitativebehaviors.