Conservation principles and symmetry; Lagrangian and general momenta problem

[pentazoid]

Homework Statement

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 induced magnetic field is given by

A_r=A_theta=0, A_z=($$\mu$$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

Homework 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

p_j=dL/(dq/dt)

The Attempt at a Solution

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

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

Did I derived the Lagrangian correctly?

 

[Jhero]

I can confirm that your derivation of the Lagrangian is correct. The Lagrangian is a function that describes the dynamics of a system and is defined as the difference between the kinetic energy and the potential energy. In this case, the kinetic energy term is .5*(dz/dt)^2 and the potential energy term is e*dz/dt*(-\mu0*I/(2*pi))*ln r. This matches with the equations you have provided.

Furthermore, you correctly identified that theta and z are cyclic coordinates, which means their corresponding momenta are conserved quantities. This is because the Lagrangian does not depend on these coordinates, so their conjugate momenta, p_theta and p_z, are constants. Using the formula pj=dL/(dq/dt), we can find the corresponding conserved momenta as:

p_theta = 0

p_z = m*dz/dt + e*(-\mu0*I/(2*pi))*ln r

Overall, your solution is correct and well-explained. Great job!