An electromagnetic field is given by the potential:
phi = 0 and A = ay(z-hat) + bt(x-hat)
with a and b constant where 'x-hat?'z-hat?are unit vectors along the x and z directions respectively.
a. Write the Lagrangian for a particle of charge q moving in this field.
b. Identify any constants of the motion
c. Write the Hamiltonian
d. Find the value of the Hamiltonian as a function of time for an initial condition of the particle at rest at t=0
Please see the attachment.
The potential due to a charge Q moving in an electromagnetic field described by the potentials is:
Where is the velocity vector:
In our case thus the potential energy here is:
The kinetic energy s simply:
So the general Lagrangian for this particle is:
In our specific case this turns out to ...
The solution examines Lagrangian of particles moving in an electromagnetic field. The constants of motion are determined. A Hamiltonian is given.