Dynamical Properties of Soft Elliptical Billiard

Two-dimensional billiards can be considered as special cases of two-dimensional potentials. These potentials must be constant at the inner part of the billiard and present an abrupt variation of their values at the coordinates on the border of the billiard. Thus, the force exerted on a particle subjected to this kind of potential is null into the billiard area and is infinite at the border of the billiard. Also, the direction of the force (and thus the potential gradient) must be normal to the frontier of the billiard. In this work we obtained a soft version of a two-dimensional billiard. Differently from the hard billiard, the particle confined in the soft billiard suffers the influence of a force during a time interval greater than zero. Due to this reason, the particle trajectory is smooth at the reflections and differs from straight lines between consecutive reflections. The obtaining of the soft billiard was made considering a particle subjected to a two-dimensional potential with a parameter capable to change the values of gradient function without however alter the shape of equipotential curves. With this, we can investigate the continuous transition of the dynamics from soft two-dimensional potential to the corresponding hard two-dimensional billiard. We opt to perform this investigation considering the elliptical geometry of the equipotential curves, where the values of eccentricities are the same for each equipotential and can be controlled by a parameter in the potential expression. Using this procedure we can reveal the changes of the numerical results by varying the hardness of the border until recover the well known phase space of hard elliptical billiard. We investigate the two-dimensional space of parameters identifying the transitions order-chaos in there.