Charged Particle Motion: Lagrange–Hamilton Orbit Dynamics

Abstract

The motion of light and objects in nature follows an orbit in which the path integral of the “action” becomes extremal. Fermat’s well-known optics principle tells us that light draws an orbit such that the time required becomes minimum between fixed point A and B. For example, Snell’s law describing refraction of light in media with different refractive indexes can be derived from Fermat’s principle (Figure 4.1). Nature is governed beautifully by the variational principle.

The variational principle for the object motion is expressed by Hamilton’s principle. The complex charged particle motion, a combination of Larmor and drift motions in the confinement magnetic field, can be simplified using the above variational principle.