Standard map in magnetized relativistic systems: Fixed points and regular acceleration

Abstract

We investigate the concept of a standard map for the interaction of relativistic particles and electrostatic waves of arbitrary amplitudes, under the action of external magnetic fields. The map is adequate for physical settings where waves and particles interact impulsively, and allows for a series of analytical result to be exactly obtained. Unlike the traditional form of the standard map, the present map is nonlinear in the wave amplitude and displays a series of peculiar properties. Among these properties we discuss the relation involving fixed points of the maps and accelerator regimes.

Controlling chaos in wave-particle interactions

“…[32], we studied the regular acceleration experienced by the particle in the region of low action I < 0.20. However, it is also possible to coherently accelerate the particle in regions of higher action as can be seen in Fig.…”

“…[32], suppose a relativistic particle with charge q, mass m, and canonical momentum p moving under the combined action of a uniform magnetic field B = B 0ẑ and a stationary electrostatic wave of wave vector k, period T , and amplitude ε/2 lying along the x axis. The transversal dynamics of this system is described by the Hamiltonian,…”

Uma introdução ao controle do caos em sistemas hamiltonianos quase integráveis

“…onde T λé o período de modulação e εé a intensidade dos impulsos (perturbação) ao longo do eixo x. Desta maneira, toda vez que t = kT λ com k ∈ Z, a perturbaçãoé ativada por um instante infinitesimalmente pequeno. Deste modo, a dinâmica transversal deste sistemaé descrita pelo Hamiltoniano [18][19][20]…”

Chaotic particle heating due to an obliquely propagating wave in a magnetized plasma

“…The difficulty in deriving such maps is that while in the absence of a background magnetic field the canonical variables appear separately in different parts of the Hamiltonian-the momentum in the kinetic part only and the coordinate in the interacting part only-the presence of the magnetic field mixes the variables in the different parts * thalestr@gmail.com of the Hamiltonian, making the integration along one wave period more cumbersome. Only recently it has been shown that by using a set of appropriate canonical transformations one can derive an exact relativistic map that also includes a background magnetic field perpendicular to the direction of the wave propagation [19]. The map led to analytical estimates on the necessary conditions to generate efficient regular acceleration.…”

“…[19] by allowing the wave to propagate at an arbitrary angle with respect to the magnetic field. In this case the wave drives the particle dynamics in both the transverse and the parallel direction with respect to the magnetic field, increasing the number of degrees of freedom in the system.…”