We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.

A body moves in a rare fied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodiesC₁andC₂ℝ³such that C₁⊂ c₂⊂ and, minimize the resistance in the class of connected bodiesBsuch thatC₁⊂ B ⊂C₁. We prove that the infimum of resistance is zero; that is, there exist ”almost perfectly streamlined” bodies.

A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the body's surface, the particles are reflected elastically. Assuming that each particle hits the body at most once (the single impact condition (SIC)), the force of resistance of the body along the flow direction can be written down in a simple analytical form. The problem of minimal resistance within this model was first considered by Newton (Newton 1687 Philosophiae Naturalis Principia Mathematica) in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section , under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated in Buttazzo et al 1995 Minimum problems over sets of concave functions and related questions Math. Nachr. 173 71–89) for the wider class of bodies satisfying the SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitch's method of solving the Kakeya problem (Besicovitch 1963 The Kakeya problem Am. Math. Mon. 70 697–706). If is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as and to 20.25 as . We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double reflections allowed.

We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from non-elastic interaction of gas particles with the body. We propose another (complementary) mechanism of creating the transversal force owing to multiple collisions of particles in cavities of the body surface. We limit ourselves to the two-dimensional case of a rough disc moving through a zero-temperature medium on the plane, where reflections of the particles from the body are elastic and mutual interaction of the particles is neglected. We represent the force acting on the disc and the moment of this force as functionals depending on ‘shape of the roughness’, and determine the set of all admissible forces. The disc trajectory is determined for several simple cases. The study is made by means of billiard theory, Monge–Kantorovich optimal mass transport and by numerical methods.

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1,12], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.