Billiards and two-dimensional problems of optimal resistance

Abstract

A body moves in a medium composed of noninteracting point particles; interaction of particles with the body is absolutely elastic. It is required to find the body's shape minimizing or maximizing resistance of the medium to its motion. This is the general setting of optimal resistance problem going back to Newton. Here, we restrict ourselves to the two-dimensional problems for rotating (generally non-convex) bodies. The main results of the paper are the following. First, to any compact connected set with piecewise smooth boundary $B \subset \mathbb{R}^2$ we assign a measure $\nu_B$ on $\partial(\text{conv}B) \times [-\pi/2, \pi/2]$ generated by the billiard in $\mathbb{R}^2 \setminus B$ and characterize the set of measures $\{\nu_B \}$. Second, using this characterization, we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special Monge-Kantorovich problems.