Charged particle in an electrodynamic field The Lagrangian for a particle with mass m and charge e moving in the general electrodynamic field {E(r,t),B(r,t)} is given in Cartesian coordinates by
L(r,r˙,t)=21mr˙⋅r˙−eϕ(r,t)+er˙⋅A(r,t),
where r=(x,y,z) and {ϕ,A} are the electrodynamic potentials of field {E,B}. Show that the corresponding Hamiltonian is given by
H(r,p,t)=2m(p−eA)⋅(p−eA)+eϕ,
where p=(px,py,pz) are the generalised momenta conjugate to the coordinates (x,y,z). [Note that p is not the ordinary linear momentum of the particle.] Under what circumstances is H conserved?