Alexander Plakhov

We continue the study of the mechanical system composed of a rod (straight line) and a ball (massless point) on the plane, where the rod rotates uniformly about a fixed point and the ball reflects elastically when colliding with the rod and moves freely between successive reflections. A sliding motion of the ball along the rod is also allowed. In a previous paper it was proved that the following regimes of motion are possible: a billiard motion, a sliding motion, and the transient regimes: a billiard motion followed by sliding and sliding followed by a billiard motion. Additionally, the asymptotic behavior of the system was studied. – In this paper we conduct a more detailed study of the dynamics. It is shown, in particular, that billiard motion is asymptotically indistinguishable from sliding one; when the number of iterations is large, the billiard particle and the corresponding sliding particle practically merge, but the billiard particle can be distinguished by vibration with very high frequency and very low amplitude.

The purpose of this article is to briefly describe the recent results (including those of the author and his collaborators) obtained on the problem of least aerodynamic resistance for convex and nonconvex bodies, for the case of translation and for motion with rotation, as well as on the free motion of bodies within the framework of Newtonian aerodynamics.

We consider the problem of camouflaging for bodies with specular surface in the framework of geometric optics. The index of visibility introduced in [ Proc. A, 473 (2017), 20170147] measures the mean deviation of light rays incident on the body’s surface. We study the problem of minimal visibility index for bodies with fixed volume contained in the unit sphere. This problem is reduced to a special vector-valued problem of optimal mass transfer, which is solved partly analytically and partly numerically. This paper is a continuation of the study started in [ Russian Math. Surveys, 64 (2009), pp. 873–938], [ Proc. A, 473 (2017), 20170147], and [ Nonlinearity, 24 (2011), pp. 847–854].

We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to the massive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle. The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.

We consider a mechanical system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points. The ball is reflected elastically when colliding with the rod and moves freely between consecutive hits. A sliding motion along the rod is also allowed. We prove the existence and uniqueness of the motion with a given position and velocity at a certain time instant. We prove that only 5 kinds of motion are possible: a billiard motion; a sliding motion; a billiard motion followed by sliding; a sliding motion followed by a billiard one; and a constant motion when the ball is at the center of rotation. The asymptotic behaviors of time intervals between consecutive hits and of distances between the points of hits on the rod are determined.

The main objective of this research is to study the properties of a billiard system in an unbounded domain with moving boundary. We consider a system consisting of an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod rotates uniformly around one of its points and experiences elastic collisions with the ball. We define a mathematical model for the dynamics of such a system and write down asymptotic formulae for its motions. In particular, we determine existence and uniqueness of solutions. We find all possible grazing impacts of the ball. Besides, we demonstrate that for almost every initial condition, the ball goes to infinity exponentially fast, with the time intervals between neighboring collisions tending to zero. The approach developed in this paper is an original combination of methods of Billiards and Vibro-Impact Dynamics. It could be a base for studying more complicated systems of similar types.