2015/02/07 by Francisco Crespo, Francisco León Crespo, Gema M. Diaz–Toca +9
Mathematics · Physics and Astronomy · #70F15 #70H05 #70H06 #70H09 #Astrophysics and Star Formation Studies #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:70F15 #msc:70H05 #msc:70H06 #msc:70H09
paper · pdf · doi:10.48550/arxiv.1502.02196
24 pages, 2 figures
arxiv created 2015/02/07 · openalex publication_date 2015/02/07 · arxiv updated 2015/02/10 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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 Gröbner 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.