2. Caustics of Hamiltonian systems
In this section we consider the quasi-periodic orbits lying on invariant tori. In particular, we present the essential ideas needed for the computation of the caustics associated with these invariant tori. For further details and more explanation see Stuchi and Vieira Martins (1995).
As it is well known, the limited trajectories of an integrable Hamiltonian system in the phase space are, in general, dense on tori of dimension equal to the number of degrees of freedom of the system (see for instance Ozório de Almeida, 1988). Given a dense orbit on an invariant torus, one can verify that when the solutions of the variational equations associated with the torus are divided by the time, they tend to tangent vectors to the invariant torus when the time interval increases with respect to the initial time. In this way, we can numerically build the tangent space to the torus at each point. Computing the singular points of the projection mapping of the torus on the configuration space and taking their projections we have that these points belong to caustics of the torus.
Therefore, to compute the caustics for a particular trajectory of the Hamiltonian system with n degrees of freedom, it is enough to follow the following steps:
- numerically computing a trajectory and n linearly independent solutions of its variational equations (n vectors are sufficient to define, at each point, the tangent space to the n -torus);
- computing the time for which the determinant of the matrix formed by the n first coordinates of the variational solutions divided by goes to zero within a required accuracy;
- determining the points of the trajectory in the configuration space corresponding to these times;
- eliminating the first points, because they correspond to a transient stage.
The remaining points in the configuration space are the points of the caustics associated with the considered trajectory and if this trajectory is dense on a torus, these points correspond to the caustic of the torus.
For a problem of n degrees of freedom, the tori are surfaces limited and differentiable in the phase space of dimension . Then, their caustics are formed by some closed surfaces of dimension , which are limited, continuous, but not differentiable and with self intersections in the configuration space, that has dimension n. From the point of view of the Hamiltonian systems theory, the invariant tori are Lagrangian manifolds, and the caustics and its singularities are Lagrangian singularities.
In particular, for two degrees of freedom systems the tori are two dimensional surfaces in a four dimensional space and the caustics are formed by one or two closed curves in the configuration space. As the torus is a surface in a four dimensional space, the caustics can be much more complicated than those one gets from their projections on a plane, from a torus of dimension two in a three dimensional space (Ozório de Almeida and Hannay, 1982). The caustic can be composed by one unique curve due to the superposition of the two closed components because of the symmetries of the Hamiltonian. In the algorithm that computes the caustics, both curves correspond to the two senses in which the determinant vanishes.
Also for two degrees of freedom, there is a very simple connection between the curves that appear in the surface of section and the caustics. Note that the closed curves that appear for the quasi-periodic orbits are defined by the intersection of the invariant torus with a plane transversal to the orbits. Therefore, the points of these transversal section curves for which the tangent is perpendicular to the position axis are points of the torus associated to the caustics. Thus one can verify that the surface of section gives an idea of just the local invariant torus. Nevertheless, it is important to point out that the method of the caustics provides information on just one invariant torus, while the Poincaré surface of section is about a set of tori.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998