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Introduction to Linear Algebra with Mathematica
Glossary
Mathieu Functions
The Mathieu equation is an ordinary differential equation with real coefficients. Its standard form with parameters (a, q) is
However, Mathieu's equation and its generalisations are more important than this single application would suggest. This equation is encountered in many different issues in physics, engineering and industry, including the stability of floating ships and railroad trains, the motion of charged particles in electromagnetic Paul traps, the theory of resonant inertial sensors, and many other problem (see expositions by McLachlan and Ruby).
The Mathieu equation has two types of solutions---stable and unstable, depending only on parameters 𝑎 and q and not on the initial conditions.
It is convenient to write Mathieu equation using a slightly different notation from that in Eq. \eqref{EqMathieu.1}, a notation which will be more convenient for our further discussion, namely, as
E.Butikov showed that the transition curves of the Ince-Strutt diagram can be determined by equations:
Let us consider the one-dimensional version of this problem. The wavefunction of the electron then satisfies the one-dimensional Schrodinger equation in which the potential V(x) is a periodic function. Suppose that the period is a, so that V(x+a)=V(x). Then, after making the change of variables
the normalized wavefunction u(x) and energy parameter λ satisfy
where \( q(x+1) =q(x) . \)
We consider this Sturm--Liouville problem on the
real line with a general 1-periodic potential q(x), assumed continuous.
A specific example of such an equation is the Mathieu equation:
where k is a real paarmeter. Its solutions, called the Mathieu functions, have been studied extensively.
Next we plot Mathieu function:
- Allievi, A. and Soudack, A., Ship stability via the Mathieu equation, International Journal of Control, 1990, Volume 51, Issue 1, pp. 139--167. https://doi.org/10.1080/00207179008934054
- Butikov, Eugene I., Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu eqution, American Journal of Physics, 86, Isuue 4, 257 (2018); https://doi.org/10.1119/1.5021895
- Daniel, D.J., Exact solutions of Mathieu’s equation, Progress of Theoretical and Experimental Physics, 2020, Volume 2020, Issue 4, April 2020, 043A01, https://doi.org/10.1093/ptep/ptaa024
- Jovanoski, Z. and Robinson, G., Ship Stability and Parametric Rolling, Australasian Journal of Engineering Education, Volume 15, 2009 - Issue 2, pp. 43--50. https://doi.org/10.1080/22054952.2009.11464028
- Mathieu, É.L., Memoir on vibrations of an elliptic membrane, 1868; Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique, Journal de Mathématiques Pures et Appliquées 13, 137–203 (1868);
- McLachlan, N.W., Theory of Application of Mathieu Functions, Dover, New York, 1964.
- Paul, W., Electromagnetic traps for charged and neutral particles, Reviews of Modern Physics, 1990, Vol. 62, No. 3, pp. 531--540. http://dx.doi.org/10.1103/RevModPhys.62.531 )
- Rand, R.H., Lecture Notes in Nonlinear Vibrations (new edition). Published on-line by The Internet-First University Press (Cornell's digital repository), 2012.
- Ruby, L., Applications of the Mathieu equation, American Journal of Physics, 1995, Volume 64, Issue 1 https://doi.org/10.1119/1.18290
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