Classical Mechanics (Kibble & Berkshire) · Problem 12.19

The calculus of variations and Hamilton's principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · lagrangian-mechanics-charged-particle-wire-magnetic-field

Problem

A particle of mass

m

and charge

e

moves in the magnetic field produced by a current

I

flowing in an infinite straight wire that lies along the

z

-axis. The vector potential

\boldsymbol{A}

of the induced magnetic field is given by

A_r = A_\theta = 0, \qquad A_z = -\left(\frac{\mu_0 I}{2\pi}\right)\ln r,

where

r

\theta

z

are cylindrical polar coordinates. Find the Lagrangian of the particle. Show that

\theta

and

z

are cyclic coordinates and find the corresponding conserved momenta.