Hamiltonian Motions Under Strong Constrains

强约束下的哈密顿运动

• 批准号: 0101969
• 负责人: Chongchun Zeng
• 金额: $ 7.03万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0101969&HistoricalAwards=false
• 关键词: Hamiltonian; Motions; Under; Strong; Constrains

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.

项目重点研究强约束条件下的哈密顿ode和偏微分方程。对于给定的由动能和势能组成的哈密顿量，考虑粒子在位形空间中受限于子流形（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包含几何概念的子流形的切线束上的哈密顿系统控制的。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，该系统具有一个额外的强势，该势会惩罚到约束子流形的距离。这个想法适用于哈密顿ode和偏微分方程。虽然已经研究了在强约束势作用下运动在有限时间区间收敛到极限几何哈密顿运动的问题，但偏微分方程中的问题基本没有触及。该项目主要关注两个问题。第一个是强惩罚运动在有限和无限时间间隔上的收敛性。第二个问题是强惩罚运动的动力学与它们的极限之间的关系，即在强约束力下的结构稳定性。主题有稳定性、周期运动、同斜运动、共振等。这个问题也可以看作是均匀化或椭圆型奇异摄动。粒子在位形空间中限于子流形的运动问题在经典力学和偏微分方程中都是自然出现的。例如，在经典力学中，只要把一根刚性杆看作弹性系数大的弹性杆，问题就属于这一类。在材料科学中，也发现一些反铁磁系统在形式上收敛于以单位二维球面为目标的几何波动方程。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更为重要。