## A cylindrical Lienard-Wiechert potential calculation using multivector matrix products.

Posted by peeterjoot on May 1, 2011

# Motivation.

A while ago I worked the problem of determining the equations of motion for a chain like object [1].

This was idealized as a set of interconnected spherical pendulums. One of the aspects of that problem that I found fun was that it allowed me to use a new construct, factoring vectors into multivector matrix products, multiplied using the Geometric (Clifford) product. It seemed at the time that this made the problem tractable, whereas a traditional formulation was much less so. Later I realized that a very similar factorization was possible with matrices directly [2]. This was a bit disappointing since I was enamored by my new calculation tool, and realized that the problem could be tackled with much less learning cost if the same factorization technique was applied using plain old matrices.

I’ve now encountered a new use for this idea of factoring a vector into a product of multivector matrices. Namely, a calculation of the four vector Lienard-Wiechert potentials, given a general motion described in cylindrical coordinates. This I thought I’d try since we had a similar problem on our exam (with the motion of the charged particle additionally constrained to a circle).

# The goal of the calculation.

Our problem is to calculate

where is the location of the charged particle, is the point that the field is measured, and

# Calculating the potentials for an arbitrary cylindrical motion.

Suppose that our charged particle has the trajectory

where , and we measure the field at the point

The vector separation between the two is

Transposition does not change this at all, so the (squared) length of this vector difference is

## A motivation for a Hermitian like transposition operation.

There are a few things of note about this matrix. One of which is that it is *not* symmetric. This is a consequence of the non-commutative nature of the vector products. What we do have is a Hermitian transpose like symmetry. Observe that terms like the and the elements of the matrix are equal after all the vector products are reversed.

Using tilde to denote this reversion, we have

The fact that all the elements of this matrix, if non-scalar, have their reversed value in the transposed position, is sufficient to show that the end result is a scalar as expected. Consider a general quadratic form where the matrix has scalar and bivector grades as above, where there is reversion in all the transposed positions. That is

where , a matrix where and contains scalar and bivector grades, and , a column matrix of scalars. Then the product is

The quantity in braces is a scalar since any of the bivector grades in cancel out. Consider a similar general product of a vector after the vector has been factored into a product of matrices of multivector elements

The (squared) length of the vector is

It is clear that we want a transposition operation that includes reversal of its elements, so with a general factorization of a vector into matrices of multivectors , it’s square will be .

As with purely complex valued matrices, it is convenient to use the dagger notation, and define

where contains the reversed elements of . By extension, we can define dot and wedge products of vectors expressed as products of multivector matrices. Given , a row vector and column vector product, and , where each of the rows or columns has elements, the dot and wedge products are

In particular, if and are matrices of scalars we have

The dot product is seen as a generator of symmetric matrices, and the wedge product a generator of purely antisymmetric matrices.

## Back to the problem

Now, returning to the example above, where we want . We’ve seen that we can drop any bivector terms from the matrix, so that the squared length can be reduced as

So we have

Now consider the velocity of the charged particle. We can write this as

To compute we have to extract scalar grades of the matrix product

So the dot product is

This is the last of what we needed for the potentials, so we have

where all the time dependent terms in the potentials are evaluated at the retarded time , defined implicitly by the messy relationship

# Doing this calculation with plain old cylindrical coordinates.

It’s worth trying this same calculation without any geometric algebra to contrast it. I’d expect that the same sort of factorization could also be performed. Let’s try it

So for we really just need to multiply out two matrices

So for we have

We get the same result this way, as expected. The matrices of multivector products provide a small computational savings, since we don’t have to look up the identity, but other than that minor detail, we get the same result.

For the particle velocity we have

So the dot product is

# Reflecting on two the calculation methods.

With a learning curve to both Geometric Algebra, and overhead required for this new multivector matrix formalism, it is definitely not a clear winner as a calculation method. Having worked a couple examples now this way, the first being the N spherical pendulum problem, and now this potentials problem, I’ll keep my eye out for new opportunities. If nothing else this can be a useful private calculation tool, and the translation into more pedestrian matrix methods has been seen in both cases to not be too difficult.

# References

[1] Peeter Joot. Spherical polar pendulum for one and multiple masses (Take II) [online]. http://sites.google.com/site/peeterjoot/math2009/multiPendulumSpherical2.pdf.

[2] Peeter Joot. {Lagrangian and Euler-Lagrange equation evaluation for the spherical N-pendulum problem} [online]. http://sites.google.com/site/peeterjoot/math2009/multiPendulumSphericalMatrix.pdf.

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