Integrable Systems Related to Separation of Variables and Symmetry Reduction

Abstract

This thesis is devoted to studying three families of integrable systems using separation of variables and symmetry reduction. Our aim is to provide concrete examples of integrable systems that fall within and without the scope of the current theory of classification of integrable systems. In this process, we also discovered related systems that have yet to be classified.

The first family of integrable systems originate from the separation of variables of the free particle on $\R^3$ in prolate spheroidal coordinates. The Lie-Poisson reduction of this system by the action of the Euclidean group $E(3)$ gives the prolate spheroidal harmonics integrable system which is symplectically equivalent to the degenerate C. Neumann system on $T^*S^2$. We will prove the presence of monodromy in the prolate spheroidal harmonics integrable system and show that this system is generalised semi-toric.

Following the same strategy, we perform symmetry reduction after separation of variables of the geodesic flow on $S^3$ to obtain our second family of integrable systems. Unlike the free particle on $\R^3$, we decided to separate the geodesic flow on $S^3$ in every orthogonally separable coordinate systems namely: ellipsoidal, prolate, oblate, Lam\'e, spherical and cylindrical coordinates. This process produces a $2$-parameter family of integrable systems on the symplectic manifold $S^2\times S^2$. We show that the moduli space for this family of integrable system is the Stasheff polytope $K^4$. In this thesis we will study these systems in detail, presenting the critical points, bifurcation diagram and action map for each system.

In Chapter 4 we consider the Harmonic Lagrange top. This system is obtained by adding a quadratic term to the potential of the Lagrange top. We study the bifurcation diagram of this systems with the help of symmetry reduction and separation of variables in Euler angles.