Eur. Phys. J. D 18, 335-338 (2002)
https://doi.org/10.1140/epjd/e20020038

Oscillations of charged particles in an external magnetic field about steady motion

Department of Physics, Faculty of Sciences, University of Sfax, B.P. 802, 3018 Sfax, Tunisia

Corresponding authors: ^a mrhimi@yahoo.fr Riadh.ElBahi@fss.rnu.tn

Received: 20 October 2001
Published online: 15 March 2002

Abstract

We develop a Hamiltonian formalism that can be used to study the particle dynamics near stable equilibria. The construction of an original canonical transformation allowed us to prove the conservation of the linear momentum P₃, which permitted the expansion of the Hamiltonian about a fixed point. The definition of the rotational variable h whose Poisson algebra properties played the essential role in the diagonalization of the quadratic Hamiltonian yielding two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable near a fixed point that come to light Heisenberg's and Harmonic Oscillator equations of motion of the particles, leading thus the association of the fixed point trajectories with arbitrary trajectories in its immediate neighborhood. The present formalism succeeded to treat the problem of free-electron laser dynamics and may be applied to similar cases.