Advertisement

Abstract

The Mathieu equation in its standard form
$$\ddot x + (a - 2q\cos 2t)x = 0$$
(6.1)
is the most widely known and, in the past, most extensively treated Hill equation. In many ways this is curious since the equation eludes solution in a usable closed form; yet many investigators have sought to describe experiments in terms of a Mathieu equation, most probably only because it contains a simple sinusoid as its periodic coefficient. By association with Fourier series it may have been assumed that once solutions to the Mathieu equation had been determined, solutions to Hill equations in general would follow. Indeed, in many ways the opposite is true in the context of the methods presented in Chap. 5.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References for Chapter 6

  1. 1.
    Mc. Lachlan, N. W.: Theory and application of Mathieu functions. Clarendon: Oxford U.P. 1947. Reprinted by Dover, New York 1964Google Scholar
  2. 2.
    Ralston, A.: A first course in numerical analysis. New York: McGraw-Hill 1965zbMATHGoogle Scholar
  3. 3.
    Dawson, P. H.; Whetten, N. R.: Ion storage in three dimensional, rotationally symmetric, quadrupole fields. I. Theoretical treatment. J. Vac. Sci. Technol. 5 (1968) 1–6Google Scholar
  4. 4.
    Dawson, P. H.; Whetten, N. R.: The monopole mass spectrometer. Rev. Sci. Instrum. 39 (1968) 1417–1422CrossRefGoogle Scholar
  5. 5.
    Lever, R. F.: Computation of ion trajectories in the monopole mass spectrometer by numerical integration of Mathieu’s equation. IBM J. Res. Dev. 10 (1966) 26–39CrossRefGoogle Scholar
  6. 6.
    Richards, J. A.: Modelling parametric processes—a tutorial review. Proc. IEEE 65 (1977) 1549–1557CrossRefGoogle Scholar
  7. 7.
    Taylor, J. H.; Narendra, K. S.: Stability regions for the damped Mathieu equation. SI AM J. Appl. Math. 17 (1969) 343–352CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Narendra, K. S.; Taylor, J. H.: Frequency domain criteria for absolute stability. New York: Academic Press 1973zbMATHGoogle Scholar
  9. 9.
    Gunderson, H.; Rigas, H.; van Vleck, F. S.: A technique for determining stability regions for the damped Mathieu equation. SIAM J. Appl. Math. 26 (1974) 345–349CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1983

Authors and Affiliations

  • J. A. Richards
    • 1
  1. 1.School of Electrical Engineering and Computer ScienceUniversity of New South WalesKensingtonAustralia

Personalised recommendations