Floquet Theory - Floquet's Theorem Applied To Mathieu Equation

Floquet's Theorem Applied To Mathieu Equation

Mathieu's equation is related to the wave equation for the elliptic cylinder.

Given, the Mathieu equation is given by

The Mathieu equation is a linear second-order differential equation with periodic coefficients.

One of the most powerful results of Mathieu's functions is the Floquet's Theorem . It states that periodic solutions of Mathieu equation for any pair (a, q) can be expressed in the form

or

where is a constant depending on a and q and P(.) is -periodic in w.

The constant is called the characteristic exponent.

If is an integer, then and are linear dependent solutions. Furthermore,

for the solution or, respectively.

We assume that the pair (a, q) is such that so that the solution is bounded on the real axis. General solution of Mathieu's equation (, non-integer) is the form

where and are arbitrary constants.

All bounded solutions --those of fractional as well as integral order-- are described by an infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency.

Another very important property of Mathieu's functions is the orthogonality :

If and are simple roots of

then:

i.e.,

where <.,.> denotes an inner product defined from 0 to π.