The reachable regions construction for the inelastic anisotropic billiards

Abstract

Modern methods of the ballistic design of space flights (entry of a spacecraft with non-zero aerodynamic quality into the planet’s atmosphere, carrying out gravity assist maneuvers around planets) are associated with the need to calculate a lot of trajectories (i.e. of the phase beams). For their effective use and determination of dynamically admissible reachability regions, it is necessary to identify and study the structure of accompanying singularities of the phase parameters and to construct the corresponding adequate and adaptive models. In this sense, the analogy of the motion of these systems with the singular motion of billiard systems turns out to be very fruitful. First of all, we are talking about anisotropic billiards with dissipation taken into account. In this work, we set and described the model problems about the flat stones skipping on the surface of the water (the ricochets) and an inelastic ball analogue which is bouncing or rolling on an imperfect surface in the presence of a side deviation effect. The main indicatrices types classification for the singular reflection from the plane surface is carried out. Model problems of the ricocheting pebble on the surface of the water surface and the inelastic ball bouncing or rolling on an imperfect surface in the presence of a side bouncing effect also the attainability region problem of the atmospheric entry are posed and described. The configurations of their maximum reachable regions are described