Poisson and symplectic reductions of 4-DOF isotropic oscillators. The van der Waals system as benchmark

TL;DR

This paper compares Poisson and symplectic reduction methods for 4-degree-of-freedom isotropic oscillators, illustrating their differences through the van der Waals system and deriving explicit normalized Hamiltonians.

Contribution

It introduces a detailed comparison of Poisson and symplectic normal form computations for 4-DOF resonant oscillators, including explicit Hamiltonian expressions and analysis of stationary solutions.

Findings

Explicit second-order Delaunay-normalized Hamiltonian derived.

Differences in stationary solutions between approaches analyzed.

Methodological pros and cons discussed for various parameter values.

Abstract

This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1 resonance with symmetries, which include several models of perturbed Keplerian systems. Normal forms are computed in Poisson and symplectic formalisms, by mean of invariants and Lie-transforms respectively. The first procedure relies on the quadratic invariants associated to the symmetries, and is carried out using Gr\"obner bases. In the symplectic approach, hinging on the maximally superintegrable character of the isotropic oscillator, the normal form is computed {\it a la} Delaunay, using a generalization of those variables for 4-DOF systems. Due to the symmetries of the system, isolated as well as circles of stationary points and invariant tori should be expected. These solutions manifest themselves rather differently in both approaches, due to the constraints among the invariants versus the singularities associated to the Delaunay chart. Taking the generalized van der Waals family as a benchmark, the explicit expression of the Delaunay normalized Hamiltonian up to the second order is presented, showing that it may be extended to higher orders in a straightforward way. The search for the relative equilibria is used for comparison of their main features of both treatments. The pros and cons are given in detail for some values of the parameter and the integrals.