Symplectic classification of coupled angular momenta

TL;DR

This paper completes the symplectic classification of coupled angular momenta systems by calculating all key invariants across the entire parameter space, including near bifurcation points, advancing the understanding of semitoric systems.

Contribution

It provides a full calculation of all symplectic invariants for coupled angular momenta, extending previous partial results to the entire parameter family.

Findings

Calculated polygon, height, and twisting-index invariants for all parameters.

Analyzed the dependence of Taylor series coefficients on system parameters.

Studied the behavior near the Hopf bifurcation where the focus-focus point degenerates.

Abstract

The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo & Vu Ngoc have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values. In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus-focus point becomes degenerate.