Do repeated roots of equations occur in the real world?

I'm wondering if repeated roots occur much in reality, outside mathematical analysis? Got any real examples from chemistry, physics, astronomy, biology, cryptology etc?





A repeated root (or "multiple root") of a polynomial equation happens when some root r has a multiplicity m%26gt;1 :


f(x) = (x-r)^m * [rest of the factors]


This complicates analysis a little, and adds special-cases.





[Yes, I know many equations of interest from the real world are not polynomials to start with anyway. But let's talk about those which are, or can be approximated within our region of interest.]





[I can only think of one example: in electronic filter or amplifier design, where we deliberately manipulate poles and zeros in a transfer function to get a desired frequency response. But strictly that is synthesis of a function which we designed, not analysis of a naturally-occurring function.]|||I can think of two off the top of my head which are physically completely different, but mathematically identical


The solution of certain differential equations reduces to a problem of solving polynomials (they are called linear differential equations with constant coefficients)


the differential equations for solving spring systems called damped motion, such as a car oscillating up and down on it's springs with its shock absorbers as the damper


if the system is under damped the car will bounce up and down to much


if it is over damped the car will not bounce enough so it will give a very rough ride


ultimately you want a situation where the spring/damper system is 'critically damped' so the ride of the car will be perfect, this is the situation where repeated roots occur in the polynomial for solving the differential equation.





the other case is in electrical circuits which contain inductors, resistors and capacitors, two examples of their use is in transformers and producing radio signals or filtering out radio interference. In the same sense as above the inductor acts like the spring, the resistor like the damper and the capacitor is like the inertia of the car, mathematically the two situations are identical so in the critically damped situation repeated roots occur in the polynomial for solving the differential equation also. Depending on the application of the inductance circuit you may or may not want the system to be critically damped though.


I'm sure there are heaps of other real life applications, which involve repeated roots, but that is all I can think of at the moment.





the end


.|||Im a theoretician. I dont care about practicality... Im beyond that :-P. Im sure examples exist in reality. Why wouldnt they? If you can construct a real life example of any polynomial, why couldnt you do the same for repeated root polynomials.





Let me tell you this. Imaginary Numbers. Complex Numbers. They dont actually exist... do they? Youd be surprised at how practical they are in physics, electronics, etc. If polynomials can have imaginary roots that have practical meaning, why couldnt repeated roots have meaning?





Instead of asking about the root themselves... ask "What happens at a root?" f(x) = x^4 and f(x) = x^2 have the same root. And the functions, themselves, look similar. But they arent. f(x) = x^4 has four identical roots... and the function acts differently than f(x) = x^2, which has only two identical roots. Its not the roots that matter... but the construction of the function, itself, that matters. The roots are just a consequence.