Abstract
The dynamics of a low-energy charged particle in an axis-symmetric magnetic field is known to be a regular superposition of periodic—although possibly incommensurate—motions. The projection of the particle orbit along the two non-ignorable coordinates (x,y) may be expressed in terms of each other: y=y(x), yet—to our knowledge—such a functional relation has never been directly produced in literature, but only by way of a detour: first, equations of motion are solved, yielding x=x(t),y=y(t), and then one of the two relations is inverted, x(t)→t(x). In this paper, we present a closed-form functional relation which allows us to express coordinates of the particle’s orbit without the need to pass through the hourly law of motion.
Highlights
IntroductionIt is fundamental to have knowledge of the particles’
In magnetized plasma research, it is fundamental to have knowledge of the particles’trajectories moving within the magnetic fields; both in natural systems, and even more in laboratory systems
Adiabatic invariants in a Hamiltonian system are phase space quantities, closely related to perturbed KAM tori, that stay constant while the system parameters are changed slowly
Summary
It is fundamental to have knowledge of the particles’. Adiabatic invariants in a Hamiltonian system are phase space quantities, closely related to perturbed KAM tori, that stay constant while the system parameters are changed slowly. They were first introduced in the old quantum theory in order to decide which dynamical variables should be quantized. Endowed with the three conserved quantities K, Pφ , μ, we may claim that a charged particle in stationary axis-symmetric magnetic fields does follow regular orbits. Define v R , v Z in terms of new parameters (w, θ ):
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