Violation of adiabaticity in magnetic billiards due to separatrix crossings

Abstract

We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of the adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance.

On Near Integrability of Some Impact Systems

“…Another approach utilized the generalized adiabatic theory in 1.5 d.o.f. systems, where the Hamiltonian dynamics are in one dimension and the boundary is slowly oscillating [12,13,25,3] . Similar approaches were employed in the study of magnetic billiards [29,6,4,5].…”

On mechanisms of destruction of adiabatic invariance in slow–fast Hamiltonian systems

“…However, this adiabatic invariant undergoes a change (a jump) ∼ ε 3/2 at passing through a narrow neighbourhood of the switching point. There is an asymptotic formula for this jump [6]. The value of this jump depends on the position of the particle on the Larmor circle (a phase) at the moment when this circle touches the boundary of the billiard.…”

On the structure of Hamiltonian impact systems

“…Indeed, for a one d.o.f. system with impacts, for all level sets that do not contain a fixed point in the domain interior, action angle coordinates (nonsmooth only at the tangency and at the singular level sets of the Hamiltonian) may be naturally defined, with the action defined as the phase space area in the billiard domain (see [2,13]).…”

“…Consider the impact system defined by eq. (1,2,3,5), where the potentials V i and the wall (Q(q 2 ) = const) are C r+1 smooth and V i are bounded from below and increase to infinity with |q i |. Proof.…”