Keywords: Lagrangian, Hamiltonian, relativistic electrodynamics, particle accelerators
Relativistic electrodynamics Lagrangian and Hamiltonian for particle accelerators
A Lagrangian L of a dynamical system is a function that summarises the dynamics of the system (Goldstein et al., 2002). If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by its direct substitution into the Euler-Lagrange equation. One important advantage of the Lagrange formulation of dynamical systems is that the formulation is not tied to any particular coordinate system - rather, any convenient set of variables may be used to describe the system. Finding the Lagrangian for a system is a mix of science and art. In the following paper we will demonstrate how to find it for the case of relativistic electrodynamics as a direct application for particle accelerators. We will show how we can start from the expression of the Lagrangian in classical electrodynamics in finding its expression for relativistic cases.