<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 Gröner bases. In the symplectic approach, hinging on the maximally superintegrable character of the isotropic oscillator, the normal form is computed <italic>a la</italic> 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.</p><p>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.</p></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 Gröner bases. In the symplectic approach, hinging on the maximally superintegrable character of the isotropic oscillator, the normal form is computed 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.

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

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1 resonance with symmetries, which include several models of perturbed Keplerian systems....

{,}	M	N	Z	S	K
M	0	4KS	0	−4KN	0
N	−4KS	0	−4L₁S	−4(KM – L₁Z)	4S
Z	0	4L₁S	0	−4L₁N	0
S	4KN	4(KM – L₁Z)	4L₁N	0	−4N
K	0	−4S	0	4N	0
L₁	0	0	0	0	0

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

Published Online: Oct 13, 2016

Page range: 473 - 492

Received: Mar 12, 2016

Accepted: Oct 10, 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00038

Keywords
Hamiltonian systems, isotropic oscillator, normal form, singular reduction, relative equilibria

© 2016 F. Crespo, G. Díaz–Toca, S. Ferrer, M. Lara, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Poisson structure in (M, N, Z, S, K, L1) invariants

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

Published Online: Oct 13, 2016

Page range: 473 - 492

Received: Mar 12, 2016

Accepted: Oct 10, 2016

DOI: https://doi.org/10.21042/AMNS.2016.2.00038

KeywordsHamiltonian systems, isotropic oscillator, normal form, singular reduction, relative equilibria

© 2016 F. Crespo, G. Díaz–Toca, S. Ferrer, M. Lara, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Fig. 2

Poisson structure in (M, N, Z, S, K, L1) invariants

Keywords
Hamiltonian systems, isotropic oscillator, normal form, singular reduction, relative equilibria