Hamiltonian group actions and dynamical systems of calogero type
Hamiltonian group actions and dynamical systems of calogero type
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DOI：
10.1002/cpa.3160310405
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发表时间：
1978-07
影响因子：
3
通讯作者：
D. Kazhdan;B. Kostant;S. Sternberg
中科院分区：
文献类型：
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作者：
D. Kazhdan;B. Kostant;S. Sternberg
近年来，人们对完全可积的Hamilton系统重新产生了兴趣，特别是与某些非线性偏微分方程如Korteweg-de弗里斯方程及其“孤子”解的研究相结合。例如，参见Moser的论文[12]，其中几个完全可积系统通过将这些问题与等谱族（即具有相同特征值的矩阵曲线）联系起来的“Lax方法”进行了研究。在本文中，我们希望建立完全可积的某些系统的方法，这是更直接地关系到群论和熟悉的程序在经典力学。我们心目中的过程是“商出”与已知积分相关的变量，以降低系统的阶数。例如，从R3中的一个总线性动量守恒的多粒子系统开始，通过观察由具有确定的总线性动量值的所有状态组成的相空间的子流形，然后忽略系统质心的位置，可以得到一个少三个自由度的系统。在物理学语言中，人们通常把这个过程称为引入“相对于质心的坐标”。在更多的数学术语中，我们可以将该过程表达如下：令“I”表示所讨论的子流形。然后相空间的辛形式对“I”的限制是奇异的，并且在“I”的每个点处具有三维零空间。这定义了“I”的三维叶理;这个叶理的商空间是辛流形，并且原始哈密顿量定义了这个商辛流形上的哈密顿量。在通常的应用中，应用这种简化方法可以简化运动方程.然而，可以想象的是，情况可能正好相反，即原始的运动方程是相当透明的，但商系统的运动方程似乎更复杂。事实上，我们将表明，Calogero系统（cf。[3])n个粒子在平方反比势下运动的直线上，以及相应的萨瑟兰系统（参见。[15]在sin-* 势下运动的圆上的n个粒子中的>是作为看起来简单得多的力学系统的替代物而出现的力学系统的例子。
In recent years there has been a renewed interest in completely integrable Hamiltonian systems, particularly in conjunction with the study of certain non-linear partial differential equations such as the Korteweg-de Vries equation and their “soliton” solutions. For example, see the paper by Moser [12] where several of these completely integrable systems are studied by the “Lax method” of relating these problems to isospectral families, that is, to curves of matrices with the same eigenvalues. In this paper we wish to establish the complete integrability of certain of these systems by a method which is more directly related to group theory and to a familiar procedure in classical mechanics. The procedure that we have in mind is the one of “quotienting out” variables associated to known integrals so as to reduce the order of the system. For example, starting out with a system of several particles in R3 for which the total linear momentum is conserved, one obtains a system with three fewer degrees of freedom by looking at the submanifold of phase space consisting of all states with a definite value of the total linear momentum and then ignoring the position of the center of mass of the system. In physical language, one usually denotes this process as introducing “co-ordinates relative to the center of mass”. In more mathematical terms, we can express the procedure as follows: let “I ‘denote the submanifold in question. Then the restriction of the symplectic form of phase space to “I’is singular and has a three-demensional null space at each point of “I ‘This defines a three-dimensional foliation of “I’; the quotient space of this foliation is a symplectic manifold and the original Hamiltonian defines a Hamiltonian on this quotient symplectic manifold. Now, in the usual applications, applying this method of reduction simplifies the equations of motion. However, it is conceivable that quite the reverse might be the case-that the original equations of motion are quite transparent, but the equations of motion of the quotient system appear more complicated. Indeed, we shall show that the Calogero system (cf.[3]) of n particles on the line moving under the inverse square potential, and the corresponding Sutherland system (cf.[15]> of n particles on the circle moving under the sin-* potential are examples of mechanical systems which arise as quotients of much simpler looking mechanical systems.