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Conservation principles and symmetry; Lagrangian and general momenta problem

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m and charge e moves in the magnetic field produced by a current I flowing in an infinte straight wire that lies along the z-axis. The vector potential induced magnetic field is given by

    A_r=A_theta=0, A_z=([tex]\mu[/tex]0*I/2*pi)*ln r, where r , theta, and z are sylindrical coordinates. Find the Lagrangian of the particle. Show that theta and z are cyclic coordinates and find the corresponding conserved momenta

    2. Relevant equations

    U=e*phi(r)-e*dr/dt*A(r)

    L=1/2*m*dr/dt*dr/dt-e*phi(r)+e*dr/dt*A(r)

    d/dt*(dT/(dq/dt))-dT/dq=d/dt(dV/(dq/dt))-dV/dq

    pj=dL/(dq/dt)

    3. The attempt at a solution
    U=e*phi(r)-e*dr/dt*A(r)

    L=.5*(dz/dt)^2+e*dz/dt*(-[tex]\mu[/tex]0*I/(2*pi))*ln r

    Did I derived the Lagrangian correctly?
     
  2. jcsd
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