## PHY450H1S. Relativistic Electrodynamics Lecture 10 (Taught by Prof. Erich Poppitz). Lorentz force equation energy term, and four vector formulation of the Lorentz force equation.

Posted by peeterjoot on February 8, 2011

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# Reading.

Covering chapter 3 material from the text [1].

Covering lecture notes pp. 74-83: gauge transformations in 3-vector language (74); energy of a relativistic particle in EM field (75); variational principle and equation of motion in 4-vector form (76-77); the field strength tensor (78-80); the fourth equation of motion (81)

# What is the significance to the gauge invariance of the action?

We had argued that under a gauge transformation

the action for a particle changes by a boundary term

Because changes by a boundary term only, variation problem is not affected. The extremal trajectories are then the same, hence the EOM are the same.

## A less high brow demonstration.

With our four potential split into space and time components

the lower index representation of the same vector is

Our gauge transformation is then

or

Now observe how the electric and magnetic fields are transformed

Sufficient continuity of is assumed, allowing commutation of the space and time derivatives, and we are left with just

For the magnetic field we have

Again with continuity assumptions, , and we are left with just . The electromagnetic fields (as opposed to potentials) do not change under gauge transformations.

We conclude that the description is hugely redundant, but despite that, local and can only be written in terms of the potentials .

## Energy term of the Lorentz force. Three vector approach.

With the Lagrangian for the particle given by

we define the energy as

This is not necessarily a conserved quantity, but we define it as the energy anyways (we don’t really have a Hamiltonian when the fields are time dependent). Associated with this quantity is the general relationship

and when the Lagrangian is invariant with respect to time translation the energy will be a conserved quantity (and also the Hamiltonian).

Our canonical momentum is

So our energy is

Or

The contribution of to the energy comes from the free (kinetic) particle portion of the Lagrangian , and we identify the remainder as a potential energy

For the kinetic portion we can also show that we have

To show this observe that we have

We also have

Utilizing the Lorentz force equation, we have

and are able to assemble the above, and find that we have

# Four vector Lorentz force

Using our action can be rewritten

is a worldline , ,

We want (to linear order in )

The variation of our proper length is

Observe that for the numerator we have

\paragraph{TIP:} If this goes too quick, or there is any disbelief, write these all out explicitly as and compute it that way.

For the four vector potential our variation is

(i.e. By chain rule)

Completing the proper length variations above we have

We are now ready to assemble results and do the integration by parts

Our variation at the endpoints is zero , killing the non-integral terms

Observe that our differential can also be expanded by chain rule

which simplifies the variation further

Since this is true for all variations , which is arbitrary, the interior part is zero everywhere in the trajectory. The antisymmetric portion, a rank 2 4-tensor is called the electromagnetic field strength tensor, and written

In matrix form this is

In terms of the field strength tensor our Lorentz force equation takes the form

# References

[1] L.D. Landau and E.M. Lifshits. *The classical theory of fields*. Butterworth-Heinemann, 1980.

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