Superintegrability in a two-dimensional space of nonconstant curvature
Superintegrability in a two-dimensional space of nonconstant curvature
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DOI：
10.1063/1.1429322
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发表时间：
2001-08
影响因子：
1.3
通讯作者：
E. G. Kalnins;J. Kress;P. Winternitz
中科院分区：
文献类型：
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作者：
E. G. Kalnins;J. Kress;P. Winternitz
如果一个具有两个自由度的哈密顿函数允许运动的三个功能无关的积分，那么它就是超可积的。这一性质在曲率恒定（可能为零）的二维空间中得到了广泛的研究，当所有的独立积分在正则动量中要么是二次的，要么是线性的。本文首先讨论了二维任意曲面流形上的超可积性问题。这是通过详细检查G. Koenigs发现的一个革命空间来完成的。我们确定了本质上有三个不同的势，当它们加到这个空间的自由哈密顿量上时就具有这种类型的超可积性。讨论了相关哈密顿-雅可比方程和薛定谔方程的分离问题。确定了与这些势相关联的经典代数和量子二次代数。
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrodinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.