This article is cited in 1 paper

Alexey Borisov Memorial Volume

The Harmonic Lagrange Top and the Confluent Heun Equation

Sean R. Dawson, Holger R. Dullin, Diana M.H. Nguyen

School of Mathematics and Statistics, University of Sydney, 2006 New South Wales, Australia

Abstract: The harmonic Lagrange top is the Lagrange top plus a quadratic (harmonic) potential term. We describe the top in the space fixed frame using a global description with a Poisson structure on $T^*S^3$. This global description naturally leads to a rational parametrisation of the set of critical values of the energy-momentum map. We show that there are 4 different topological types for generic parameter values. The quantum mechanics of the harmonic Lagrange top is described by the most general confluent Heun equation (also known as the generalised spheroidal wave equation). We derive formulas for an infinite pentadiagonal symmetric matrix representing the Hamiltonian from which the spectrum is computed.

Keywords: symmetric rigid body, Lagrange top, Hamiltonian Hopf bifurcation, quantisation, confluent Heun equation.

MSC: 70E17, 81Q99

Received: 01.11.2021
Accepted: 13.06.2022

Language: English

DOI: 10.1134/S1560354722040049

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