Semiclassical Monodromy and the Spherical Pendulum as a Complex Hamiltonian System

Alber, M. S. and J. E. Marsden

Fields Inst. Commun., 8, (1996), 1-18

Abstract:

In the present paper, we apply our approach to effects that are related to the problem of semiclassical monodromy. A crucial point in doing this is to consider the complexification of the system. In particular, we describe new complex angle representations and Hamiltonians for the classical simple spherical pendulum (out angle representations hold for the n-dimensional case and are different from those of the above mentioned authors even the the real case, and are based on the Abel-Jacobi map). In particular, this yields new exponential Hamiltonians and angle representations on homoclinic varieties and leads to the introduction of Maslov indices of closed curves in Lagrangian submanifolds of the cotangent bundle of the configuration space. These Lagrangian submanifolds are defined by the first integrals of the problem and will be described below. Then we develop complex geometric asymptotics with corresponding quantum conditions. These quantum conditions include classical and complex monodromy together with phase shifts that are related to Maslov indices after transporting the system along certain closed curves in the space of parameters. We refer to those types of phase shifts which are associated to the quantum conditions and also phase shifts that are associated to singularities in the space of parameters as semiclassical monodromy.