Three unequal masses on a ring and soft triangular billiards

Collisions with hard (infinite) walls in billiards systems are usually described by instantaneous reversal of the particles linear momentum. From this, simple analytical relations of velocities and angles before and after the collisions with the walls are obtained. However, in order to analyze the transition to soft walls, which are more realistic, it is essential to have well defined equations of motion since, in general, no simple analytical solutions are obtained. The present work suggests that appropriated soft walls potentials are those for which the corresponding forces become ``delta functions" in the limit of hard walls. This allows for better numerical investigation of the soft-hard transition. A general scaled Hamiltonian is derived for three unequal masses interacting particles on a frictionless ring, which nicely describes the transition and shows that the dynamics occurs inside a soft triangular billiard. The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.