Fundamentals of Plasma Physics 2025, 20 - Motion of Single Plasma Particles
As the theory comes from the Vlasov equation, a complementary formulation to Hamilton-Lagrange originating from electromagnetic quantities. These are the Lorentz equations. The Hamiltonian approach is as we're accustomed to, with all the usual benefits, resulting in a Lagrangian for a charged particle in an EM-field, which generates the Lorentz eq. Without symmetry, the canonical momentum becomes P = mv + qA(x, t).
The question arises whether the result is robust against breakdowns of symmetry. For adiabatic plasma, a pendulum equation of motion can be used as a test potential, which should be solved using WKB method. This remains solvable and while the frequency is time-dependent, the pendulum amplitude is not constant, meaning energy is not conserved. Instead, the action integral over the period of oscillation is conserved. The action is an area in phase-space enclosed by the tranjectory and shows that for a slowly changing pendulum frequency, and is a constant of motion. This transfers to any dynamical systems with equations of motion of this form. A general adiabatic invariant can be achieved through extension of the WKB method leads to an action of the form S = λ2∫02π dϕ cos2ϕ = const.
Drift velocities emerge from the Lorentz equation. As the magnetic force acts perpendicular to drift always, it does no work initially, leaving only effects of uniform electric fields. When assumed uniform and static, the resulting motion is a simple "falling into the field origin" motion. As a particle begins falling, the magnetic field begins affecting it, curving its trajectory. The velocity decelerates as it's trajectory is curved back completely, and eventually reverses. The process repeats, and so the particle travels in approximate semicircles. Averaging over that gives the drift velocity. as vF = (E×B)/(qB2). If the force acting on the particle at initial condition is one that changes "slowly and continuously", then the change (3-dimensional derivative) create different forms of drifts. The total drift is the sum of all these drifts.
