The Lagrangian and Hamiltonian for a Charged Particle in an EM Field

Here I go over the Lagrangian and Hamiltonian for a charged particle in an electromagnetic field. The first part is a justification of the Lagrangian for those who have had it presented to them with no motivation. This justification comes in the form of verification that EoMs from the Euler Lagrange equations recover the familiar form of the Lorentz force. Following this, the conversion to the Hamiltonian is shown.