On Totally integrable magnetic billiards on constant curvature surface

Hyperbolic Magnetic Billiards on Surfaces of Constant Curvature

We consider classical billiards on surfaces of constant curvature, where the charged billiard ball is exposed to a homogeneous, stationary magnetic field perpendicular to the surface. We establish sufficient conditions for hyperbolicity of the billiard dynamics, and give lower estimation for the Lyapunov exponent. This extends our recent results for non-magnetic billiards on surfaces of constan...

Hyperbolic Billiards on Surfaces of Constant Curvature

We establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature. This extends known results for planar billiards. Using these conditions, we construct large classes of billiard tables with positive Lyapunov exponents on the sphere and on the hyperbolic plane.

On algebraically integrable outer billiards

We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse. In this note, an outer billiard table is a compact convex domain in the plane bounded by an oval (closed smooth strictly convex curve) C. Pick a point x outside of C. There are two tangent lines from x to C; choose one of them, say, the right one...

Constant mean curvature surfaces and integrable equations

Integrable Systems and Metrics of Constant Curvature

PACS : 02.30.J, 11.10.E; MSC : 35L65, 35L70, 35Q35, 58F05, 58F07 keywords: Lagrangian, metrics of constant curvature, Hamiltonian structure, reciprocal transformation, Poisson brackets. Abstract. In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with me...