Holger R Dullin | Searcharxiv

arXiv2023

The Lagrange top and the fifth Painlevé equation

We show that the Lagrange top with a linearly time-dependent moment of inertia is equivalent to the degenerate fifth Painlevé equation. More generally we show that the harmonic Lagrange top (the ordinary Lagrange top with a quadratic term added in the potential) is equivalent to the fifth Painlevé equation when the potential is made time-dependent in an appropriate way. Through this identification two of the parameters of the fifth Painlevé equation acquire the interpretation of global action variables. We discuss the relation to the confluent Heun equation, which is the Schrödinger equation of the Lagrange top, and discuss the dynamics of $P_V$ from the point of view of the Lagrange top.

nlin.SI↗

arXiv2013

The Vanishing Twist in the Restricted Three Body Problem

This paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three-body problem. Near the Lagrangian equilibrium $\mathcal{L}_4$ a twistless torus is created near the tripling bifurcation of the short period family. Decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3/10 and 2/7. This scenario is studied by numerically integrating the regularised Hamiltonian flow, and finding rotation numbers of invariant curves in a two-dimensional Poincaré map. To corroborate the numerical results the Birkhoff normal form at $\mathcal{L}_4$ is calculated to eighth order. Truncating at this order gives an integrable system, and the rotation numbers obtained from the Birkhoff normal form agree well with the numerical results. A global overview for the mass ratio $μ\in (μ_4, μ_3)$ is presented by showing lines of constant energy and constant rotation number in action space.

math-ph↗