Linear stability in billiards with potential

Abstract

A general formula for the linearized Poincaré map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this reduces to well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.

On Near Integrability of Some Impact Systems

“…Taking ε < ε 0 (H, δ/2, ρ/2), insures that if I 0 ∈ S g (H, δ, ρ) then it is at least ∆ away from the boundary of S g (H, δ/2, ρ/2), where ∆ = min(ρ/4, K 1 δ, K 2 δ| ln(2δ)|) and K 1,2 (H) are some constants depending on the unperturbed rotation rates (see Eq. 7,11). It follows that the map (14) The destroyed, resonant tori correspond to rational values of the modified rotation number…”

“…Here we provide such a class of prototype impact systems which are near-integrable and are amenable to analysis. Previous near-integrability results for HIS have utilized the local dynamics near periodic orbits [11,4,5,17], near a smooth convex boundary [31,6,5] and near saddle-center homoclinic connection of a quadratic potential with impacts [20]. Another approach utilized the generalized adiabatic theory in 1.5 d.o.f.…”

Smooth Hamiltonian Systems with Soft Impacts

“…The dynamics of such systems is non-trivial even in the one-dimensional case [6]; in higher dimensions, only partial results exist. An extensive theoretical investigation of such systems is presented in [5].Notably, all of the applications mentioned above involve a steep repulsion term, which is replaced in these works by a hard-wall potential for simplicity. Here, we provide conditions under which this approximation is justified, and examples in which it is utilized as a computational tool via continuation methods.…”

“…The dynamics of such systems is non-trivial even in the one-dimensional case [6]; in higher dimensions, only partial results exist. An extensive theoretical investigation of such systems is presented in [5].…”

“…Finally, Theorem 1 could possibly be extended to other important cases; for example, cases in which the potential and the billiard boundary move in time and cases in which the kinetic energy depends on the position, as when the particles are charges and are subjected to a magnetic field (see [1,2,5]). Such extension will further enhance the applicability of this methodology to additional fields of physics and chemistry.…”