Hamiltonian Motions Under Strong Constrains

该项目的重点是在强约束力作用下的哈密顿常微分方程和偏微分方程组。对于由动能和势能组成的给定哈密顿量，考虑限制在位形空间中的子流形(与时间无关)的粒子，即完整约束。约束粒子运动的理想化由定义在涉及几何概念的该子流形的切丛上的哈密顿系统支配。在物理学中，另一种实现约束的方法是考虑原始空间中的系统，该系统具有额外的强势，这将惩罚到约束子流形的距离。这一思想既适用于哈密顿微分方程组，也适用于偏微分方程组。虽然常微分方程组已经研究了强约束势下的运动收敛到有限时间区间上的极限几何哈密顿运动，但偏微分方程组中的问题基本未被触及。该项目关注两个问题。第一个是强惩罚运动在有限和无限时间间隔上的收敛。第二个问题是强约束运动的动力学与其极限的关系，即在强约束力作用下的结构稳定性。问题的主题是稳定性、周期运动、同宿运动、共振等，也可以看作是齐次化或椭圆型奇异摄动问题。质点在位形空间中的运动限制在子流形上的问题在经典力学和偏微分方程组中都是自然出现的。例如，在经典力学中，当刚性杆被认为是具有大弹性系数的弹性杆时，问题就属于这一类。在材料科学中还发现，一些反铁磁系统形式上收敛到单位二维球面上的几何波动方程。因此，研究约束运动如何收敛具有重要意义。此外，由于惩罚运动具有高频振荡，因此更重要的是研究渐近定性行为之间的关系。

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.