Interesting Idea: Showing a Force is Conservative with Complex Analysis

[Tleilaxu_Ghola]

Preface:The best way I've been taught how to prove that a force is conservative is to take the curl of the force and show that it is equal to zero. That's pretty quick, but after studying for a complex analysis midterm this idea struck my mind. I'm not a master of complex analysis, so there may be errors. Check my math, please. It's also possible that this has been pointed out before... if so please point me in the right direction.

Basic Premise: The essential condition that makes a force conservative is that it creates an energy potential which is path independant. That is to say that any contour integral of the force on a simple closed contour is zero (basic theorem of multivariable calculus).

Suppose your position is constrained to two dimensions and can be written in cartesian coordinates as: (x,y)

Now let's suppose you have a force of the form: f(x,y).

The method:

1. Transform the force to f(Z). Use the linear transformation: Z = X + iY
2. Put the force into the form:
g(z)*(z-q)^-n, such that q is some complex constant
3. Show that the (n-1)th derivative of g(z) = 0.

RESULT: You prove that the force is conservative in some neighborhood around the singularity at the point q. The radius of this neighborhood is dependant on the proximity of other singularities present in the force field. If more than one exists, then the the force is conservative only on the maximal disc between the two singular points.

Notes:
Typically you can declare your origin wherever you like and set q = 0.


Proof:
The proof is pretty simple, provided I didn't screw anything up. Basically, you just need to show that a singularity in the force field has no residue (ie residue = zero).

Here's how one proves this:

1. Cauchy's Residue Theorem: The contour integral over a simple closed contour in a multiply connected domain of some function f(z), where z is a complex number, is proportional to the sum of all the residues of the function f(z) at all its singular points. The constant of proportionality is 2*pi*i.
2. We want the contour integral of our complex force to be zero, so that means that all the residues must be zero.
3. The easiest way to find a residue is to find its poles. That's the part where I find g(z)*(z-q)^-n. This form tells us that there is a pole of order n at point q.
4. The residue is given by the equation: the (n-1)st deriviative of g(z) divided by (n-1)!. If we want all the residues to be zero, we need to show that the (n-1)st deriviative of g(z) is zero.

Essentially this is just an application of Cauchy's Residue Theorem.



Why is this special/different/any good?

You still need to be able to separate the force into X, Y, & Z components, just like you would if you were getting ready to find the curl of the force. Provided you can do that, then you just multiply the Y component by i to get the force onto the complex plane. Once you're there now all you need to do is find the singular points (usually by inspection) and use the poles to calculate the residues. (Alternatively you could use the residue definition and laurent series to find the residues).

With the curl method you just get a simple yes or no answer to whether or not the force is conservative. With this method you can find on what radius in the complex plane is the force conservative... and that radius may be finite.

I think that this should be considered to be pretty significant for the simplification of some problems. If you know that your position vector is constrained to a certain disc, you may be able to simplify your calculations if you can show that your force is conservative on that disc (but not necessarily outside of it).

Another good thing: If you can approximate your force with a laurent series you only need to worry about getting the 1/z terms (which should have a coefficient of zero for a conservative force) and some number of other terms according to the accuracy you require. I'm not sure if it is possible to use laurent series approximations with the curl method.

Thoughts?

 

[Tleilaxu_Ghola]

I'm having trouble implimenting the transformation for this. I thought I understood it in concept: make the y-axis the imaginary line and the real-line the x-axis.

Essentially that would mean: f(x,y) = f(x, iY) and then use x + iY = z, where iY = y.

That's not very easy though... is there a better way to transform the function onto the complex plane? (reducing it from two dimensions to one?)

 

[Tleilaxu_Ghola]

Okay, so this obviously isn't as interesting as I thought...

A general transformation of a function f(x,y) to f(z) is still eluding me. That's unfortunately the key to making this idea more general.

If I remember correctly, all conservative forces are central, meaning they can be written as f(r). If that's the case, then r could be traded like a dummy variable for z... and then we can apply Cauchy's Residue Theorem. Right?

I was hoping to make as little assumptions about the force-field function as possible.

Do y'all think it might be possible that a force field is conservative in some domains, but not in others? If so, then I'm hoping this method will allow one to locate those domains.