A particle of mass and charge moves in the -plane in a uniform magnetic field ,
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Hamiltonian mechanics and canonical structure
Action and Symmetry, Ch The Hamiltonian, phase space and the variational principle in phase space
Problem
A particle of mass m and charge Qe moves in the xy-plane in a uniform magnetic field B=Bz^, B=0.50T, and a uniform electric field E=Ex^, E=1.0×104Vm−1. Use the gauge A=(0,Bx,0) and the scalar potential ϕ=−Ex, so that E=−∇ϕ. The Hamiltonian is H(x,y,px,py)=2mpx2+(py−QeBx)2−QeEx.
((a)) Write Hamilton's four equations for this H and say which momentum is conserved and why.
((b)) Eliminate the momenta to get two second-order equations for x(t) and y(t). Show that a solution exists with x˙=0 and y˙ constant, and find that y˙ in terms of E and B alone.
((c)) Show that every other solution is uniform circular motion at ωc=QeB/m about a centre that moves with the velocity found in part (b). Describe the path in the xy-plane.
((d)) Evaluate the drift velocity for the stated fields, and say whether it depends on m or on Qe, including the sign of Qe.