Consider the simplest case of a single ion. The equations of motion for a single ion of mass m and charge ne in a Paul trap are:
where is the applied static potential, is the amplitude of the applied oscillating potential, is the frequency of the oscillating potential, and are the electrode dimensions (as depicted in Fig. ). Each dimension (x, y, z) of the second-order differential equation for a single ion's motion in the Paul trap can be expressed as a one-dimensional Mathieu equation:
through the transformations:
where , describe the dynamics in the radial (xy) plane and , describe axial motion. The x, y, and z-coefficients of -1, -1, and 2 in Eq. account for the relationships between , and , .
The Mathieu equation is a linear second-order differential equation with periodic coefficients. It belongs to a family of equations known as Hill's equations which can be written as:
where is written as the square of a function not to imply that the quantity , but rather to suggest analogy to the harmonic oscillator. French mathematician Mathieu investigated the Mathieu equation in 1868 while seeking a description of the vibrations of an elliptical membrane. 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.[29,28] In addition to being theoretically interesting, the Mathieu functions are applicable to a wide variety of physical phenomena--- for example, problems involving waveguides, diffraction, amplitude distortion, and vibrations in a medium with modulated density.