Mathieu’s equation

This equation is Mathieu s equation in the usual form. 

Although Hq. (152) can in principle be solved by the development of y(x) in a power series, the periodicity of the argument of cosine, namely, 2jc = Na complicates the problem. The most important application of Mathieu s equation to internal rotation in molecules is in the analysis of the microwave spectra of gases and vapors. The needed solutions to equations such as Eq. (152) are usually obtained numerically. 

In this approach, the horizontal coordinates x[t] and y[t] both satisfy linear, second order equations. The exact trajectories satisfy Lame s equation of order two, which is extremely difficult to analyse. The corresponding equation for the linear approximation is Mathieu s equation, which is known to have both periodic and aperiodic solutions. 

The corresponding initial conditions for F[r], which satisfies the same Mathieu s equation, are... 

Numerical simulations of the spherical pendulum for arbitrary values of K and W will usually reveal a very complicated, a periodic motion of the type shown in Fig. 2, but in some cases the motion is periodic. The theory can be found in Refs. [9,11], but is summarised here. Let r be any integer or simple fraction (such as 3/2, etc.). Then solutions of Mathieu s equation of the form... 

The equation of motion may be transformed and rearranged to yield Mathieu s equation, which has been studied extensively to establish its stability characteristics (see Abramowitz and Stegun, 1964) ... 

Equation (4-321) is a special case of a DE that is known as Mathieu s equation, for which the standard form is... 

Figure 4-17. Stability boundaries for solutions of Mathieu s equation (reproduced from Ref. 30). The shaded regions correspond to unstable regions in the a-b plane and the unshaded are stable regions.

We can obtain additional insight into the nature of the solutions of Mathieu s equation by extending the asymptotic solution described previously to small, but nonzero, values of e. Of particular interest is the behavior of solutions near the first resonant instability point ak = 1/4. Referring to Fig. 4-17, we see that there is a finite region around ak = 1/4 where solutions for nonzero values of ebk are predicted to be unstable. In this section, we seek an asymptotic expression in terms of e for the critical values of ak 1 /4 that separate the regions of stable and unstable solutions. Thus we suppose that... 

The basic mathematical equation of the quadrupole mass spectrometer is Mathieu s equation. 

Mathieu s equation is a linear second-order differential equation with periodic coefficients. It belongs to the family of ///// s equations. The one-dimensional Mathieu s equation is written in standard form as... 

The solutions to Mathieu s equation comprise an orthogonal set and possess the curious property that the coefficients of their Fourier series expansions are identical in magnitude, with alternating signs, to corresponding coefficients of their Bessel series expansions [2, 3]. Floquefs theorem asserts that any solution of equation (Eq. 20.7) is of the form... 

The first term represents the kinetic rotational energy. This equation is amenable to Mathieu s equation and analytical formulae for the eigenstates are known. However, with computer facilities available nowadays, it is faster to calculate numerically both eigenvalues and eigenfunctions [59]. If necessary, these calculations can be performed for potential functions including several Fourier terms. 

The solutions to Mathieu s equation are of two types, (i) periodic but unstable, and (ii) periodic and stable. Solutions of type (i) form the boundaries of unstable regions on the stability diagram and correspond to those values of a trapping parameter, j z that are integers, that is, 0, 1, 2, 3,. / z is a complex function of az and qz that is... 

---