# Lagrangian of Particle Moving in Electromagnetic Field

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

https://brainmass.com/physics/energy/lagrangian-particle-moving-electromagnetic-field-239533

#### Solution Preview

Please see the attachment.

The potential due to a charge Q moving in an electromagnetic field described by the potentials is:

(1.1)

Where is the velocity vector:

(1.2)

In our case thus the potential energy here is:

(1.3)

The kinetic energy s simply:

(1.4)

So the general Lagrangian for this particle is:

(1.5)

In our specific case this turns out to ...

#### Solution Summary

The solution examines Lagrangian of particles moving in an electromagnetic field. The constants of motion are determined. A Hamiltonian is given.