Time-independent Hamiltonians describing systems with friction: the “cyclotron with friction”

Abstract

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler-Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is more complex, and there exist systems of ODEs which cannot be described by any Lagrangian. In this paper we display Hamiltonians which describe the behaviour of a charged particle moving in a plane under the combined influence of a constant electric field (in the plane) and a constant magnetic field (orthogonal to the plane) as well as a friction force proportional to the velocity ("cyclotron with friction").

A Hamiltonian yielding damped motion in an homogeneous magnetic field: quantum treatment

“…As was pointed out in [1], the Hamiltonian (2.1a) is not rotationally invariant, even though its classical orbits have the property that rotating one orbit leads to another such orbit. In other words, rotations yield a symmetry of the equations of the motion, but not of the underlying Hamiltonian structure.…”

“…Here E, P, a and b are 4 a priori arbitrary (real) parameters; the two canonical variables x respectively y are the two Cartesian coordinates of the (charged) particle moving in the plane and p x respectively p y the corresponding canonical momenta. This Hamiltonian features the following conservation law, as described in [1]:…”

“…The final results for the Hamiltonian corresponding to θ = 0 can, however, easily be extended to the general case by performing the appropriate rotation in the position and momentum variables. For further remarks on the classical Hamiltonian, see [1].…”

“…In a previous paper [1], we displayed a time-independent Hamiltonian describing the motion of a charged particle moving in two dimensions under the combined influence of a magnetic field perpendicular to the plane of motion and a friction force proportional to the particle velocity. While, physically speaking, such a model arises via the coupling of the particle to a large number of external degrees of freedom, which are then averaged over, it was shown in [1] that it is possible to have a description via a time-independent Hamiltonian involving no additional degrees of freedom.…”

“…For a description of such Hamiltonians for the one-dimensional damped harmonic oscillator, however, see [3]. We presented-in the context of classical mechanics-the Hamiltonian model of the damped motion of a charged particle in a plane in the presence of a uniform magnetic field orthogonal to that plane in [1] and discussed its symmetry properties. Here we proceed to discuss the quantization of this model.…”