On the action for a charged particle moving in a magnetic field

TL;DR

This paper introduces a novel approach using Maupertuis principle to determine the canonical momentum of a charged particle in a magnetic field, simplifying the application of quantization conditions without relying on the traditional Lagrangian method.

Contribution

It presents a new method based on Maupertuis principle to derive canonical momentum and action for charged particles in magnetic fields, bypassing the Lagrangian approach.

Findings

Inclusion of a flux-dependent term in the action is necessary.

The method simplifies applying quantum conditions to charged particles.

It provides a straightforward way to obtain canonical momentum in electromagnetic fields.

Abstract

Application of the Bohr-Wilson-Sommerfeld quantization condition to a charged particle in a uniform magnetic field requires knowledge of the canonical momentum of such a particle, which in turn requires students to know about the vector potential. The canonical momentum in this situation is conventionally obtained by introducing an appropriate additional term involving the corresponding vector potential in the Lagrangian. In this work, we take a different approach to this problem by analyzing it with Maupertuis principle of least action. We show that satisfying this principle for a charged particle moving in a uniform magnetic field requires that a term proportional to the flux passing through the area enclosed by the trajectory of the particle be included in its action. This additional term accomplishes two tasks. First, it facilitates applying the Bohr-Wilson-Sommerfeld quantum condition for a charged particle moving in a uniform magnetic field without using the Lagrangian approach, and, secondly, it gives the appropriate canonical momentum and Lagrangian for a charged particle in a general electromagnetic field in a straightforward manner.