Hamiltonian Motions Under Strong Constrains

本项目主要研究强约束力下的Hamilton常微分方程和偏微分方程。对于给定的由动能和势能组成的哈密顿量，考虑粒子被限制在位形空间中的子流形（与时间无关），即完整约束。约束粒子运动的理想化是由定义在这个子流形的切丛上的包含几何概念的哈密顿系统所支配的。在物理学中，实现约束的另一种方法是考虑原始空间中的系统具有额外的强势，该势惩罚到约束子流形的距离。这一思想既适用于哈密顿常微分方程，也适用于偏微分方程。虽然强约束势下的运动在有限时间区间上收敛到极限几何Hamilton运动的问题已经在常微分方程中得到了研究，但在偏微分方程中的问题基本上还没有触及。该项目侧重于两个问题。第一个问题是强惩罚运动在有限和无限时间区间上的收敛性。第二个问题是强惩罚运动的动力学与其极限之间的关系，即强约束力下的结构稳定性。这些问题的主题是稳定性、周期运动、同宿运动、共振等，也可以看作是均匀化或椭圆型奇异摄动问题，而粒子在构形空间中的运动问题则是经典力学和偏微分方程中自然出现的问题。例如，在经典力学中，只要一根刚性杆被认为是弹性的，弹性系数很大，问题就属于这一类。在材料科学中还发现，某些反铁磁系统形式上收敛于二维单位球上的几何波动方程。因此，研究约束运动如何收敛是非常重要的。此外，由于惩罚运动具有高频振荡，因此研究渐近定性行为之间的关系就显得更为重要。

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.