Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Newtonian aerodynamics models a freely moving body interacting by elastic boundary reflection with a rarefied medium of infinitesimal particles. In the stated one-dimensional special case the body is a point of mass $1$ and the medium consists of identical infinitesimal particles. Here $(x,v)\in\mathbb R^2$ is particle position and velocity, $\mu_t$ their distribution measure, and $X(t),P(t)=X\prime(t)$ are the position and velocity of the massive point. The request includes formulating the dynamics, rather than assuming a preselected evolution equation.

Statement recorded in the supplied source.

A body moves freely in a rarefied medium in $\mathbb{R}^n$, $n\geq1$, under Newtonian aerodynamics. Determine the equations of motion and prove existence and uniqueness. In the one-dimensional case, formulate the motion using a measure $\mu_t$ on particle phase space together with the massive particle's position $X(t)$ and velocity $P(t)=X'(t)$. [S2]

Short English statement

Formulate the free-body dynamics in a rarefied particle medium in every positive dimension and prove existence and uniqueness, including the one-dimensional measure-valued particle model coupled to a point of mass one.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200012 (AMR-051-0012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Alexander Plakhov, Problem 3, p. 9. https://arxiv.org/pdf/2110.10750v2.