可积系统的Maslov指数与非退化奇点
The Maslov index and nondegenerate singularities of integrable systems
摘要：
考虑R^{2n}中具有运动积分F =（F_1，.，F_n）。非退化奇点是F的临界点，其中dF = n-1，并且具有确定的线性稳定性。非退化奇点的集合是一个余维2辛子流形下的流动不变。我们发现，封闭曲线的Maslov指数是一个贡献+/-2的总和，从非退化奇点，它是封闭的，该符号取决于当地的方向和稳定性的奇点。对于单自由度系统，这对应于众所周知的闭合曲线的庞加莱指数公式，即椭圆和双曲不动点之间的定向差。我们还得到了一个公式不变（n-1）维环面的非退化奇异集的Liapunov指数。例子包括旋转对称的n-自由哈密顿，而应用程序的周期户田链中描述的配套文件。
英文摘要：
We consider integrable Hamiltonian systems in R^{2n} with integrals of motion F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical points of F where rank dF = n-1 and which have definite linear stability. The set of nondegenerate singularities is a codimension-two symplectic submanifold invariant under the flow. We show that the Maslov index of a closed curve is a sum of contributions +/- 2 from the nondegenerate singularities it is encloses, the sign depending on the local orientation and stability at the singularities. For one-freedom systems this corresponds to the well-known formula for the Poincaré index of a closed curve as the oriented difference between the number of elliptic and hyperbolic fixed points enclosed. We also obtain a formula for the Liapunov exponent of invariant (n-1)-dimensional tori in the nondegenerate singular set. Examples include rotationally symmetric n-freedom Hamiltonians, while an application to the periodic Toda chain is described in a companion paper.
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