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Sebastián Ferrer

Publications and source records attributed to Sebastián Ferrer.

6 recordsLinked to original sources
arXiv2015

On the N-Extended Euler System I. Generalized Jacobi Elliptic Functions

We study the integrable system of first order differential equations $ω_i(v)'=α_i\,\prod_{j\neq i}ω_j(v)$, $(1\!\leq i, j\leq\! N)$ as an initial value problem, with real coefficients $α_i$ and initial conditions $ω_i(0)$. The analysis is based on its quadratic first integrals. For each dimension $N$, the system defines a family of functions, generically hyperelliptic functions. When $N=3$, this system generalizes the classic Euler system for the reduced flow of the free rigid body problem, thus we call it $N$-extended Euler system ($N$-EES). In this Part I the cases $N=4$ and $N=5$ are studied, generalizing Jacobi elliptic functions which are defined as a 3-EES. Taking into account the nested structure of the $N$-EES, we propose reparametrizations of the type ${\rm d}v^*=g(ω_i)\,{\rm d}v$ that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the {\sl Jacobi amplitude}. In Part II we consider geometric properties of the $N$-system and the numeric computation of the functions involved. It will be published elsewhere.

math.DS↗
arXiv2015

Expanding the simple pendulum's rotation solution in action-angle variables

Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions the transformation to action-angle variables may need to remain in implicit form. This is exactly the case of the simple pendulum, where in order to make explicit the transformation to action-angle variables one needs to resort to nontrivial expansions of special functions and series reversion. Alternatively, it is shown that the explicit expansion of the transformation to action-angle variables can be constructed directly, and that this direct construction leads naturally to the Lie transforms method, in this way avoiding the intricacies related to the traditional expansion of elliptic functions.

nlin.SI↗
arXiv2015

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

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.

math.DS↗
arXiv2010

Integration of the Euler-Poinsot Problem in New Variables

The essentially unique reduction of the Euler-Poinsot problem may be performed in different sets of variables. Action-angle variables are usually preferred because of their suitability for approaching perturbed rigid-body motion. But they are just one among the variety of sets of canonical coordinates that integrate the problem. We present an alternate set of variables that, while allowing for similar performances than action-angles in the study of perturbed problems, show an important advantage over them: Their transformation from and to Andoyer variables is given in explicit form.

nlin.SI↗
arXiv2009

Families of Canonical Transformations by Hamilton-Jacobi-Poincaré equation. Application to Rotational and Orbital Motion

The Hamilton-Jacobi equation in the sense of Poincaré, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction. We illustrate our approach dealing with orbital and attitude dynamics. Based on the use of Whittaker and Andoyer symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in orbital and attitude dynamics. In addition, new canonical transformations are demonstrated.

nlin.SI↗