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    Lagrangian of Particle Moving in Electromagnetic Field

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    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

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    Solution Preview

    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 ...

    Solution Summary

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