Dynamic Strong Focusing Force

In a dynamice strong force system, the force is a periodic function with a possible fixed contribution, i.e.,

F = - k (U - V cos wt) u

where U is a constant offset component, V is the amplitude of the oscillating component with an angular frequency w = 2pn.

To determine how the positions of a particle with mass m, will be determined by this force, we need to solve Newton's equation of motion for this system, i.e.

Thus we need to find solutions to the differential equation:

This equation has 5 constants, m, k, U, V, and w, and two variables, u and t in it. It would be wise to simplify this expression to reduce the amount of constants and variables, and at the same time come up with a form that some mathematicians may already have investigated.

Mathieu Equation

By using

it is possible to rewrite the equation of motion into a form similar as the so-called Mathieu equation.

This expression only has 2 constants, a and q, and two variables u and x, a modified time variable.

Floquet's theorem states that a solution to a linear differential equation with periodic coefficents, of which the Mathieu equation is one, can be expressed as

where f(x) is a periodic function with the same periodicity as the coefficient in the differential equation.

Since the Mathieu equation is not changed by substituting x by -x, u(-x) must be a solution too. Moreso, since u(-x) is not a constant multiple of u(x), the solution is an independent one, and the complete solution to the Mathieu equation can be expressed as a linear combination of u(x) and u(-x)

where we have two periodic functions f(x) and f(-x) and coefficients c₁ and c₂, which are constants of integration. The values of the coefficients depend on the initial conditions of the position and velocity of the ion and phase of the oscillating field. If c₁x c₂ is not zero, this kind of solution is called a Floquet solution.

The characteristic exponent, m, may be real, imaginary or complex, and can be expressed as m = a + ib. The value of m depends a and q, and gives rise to distinct sets of solutions that can be characterized as either

stable: the solution is finite at all times, or
unstable: the solution will tend to inifinity as x increases

These possibilities are

a <> 0: with a > 0 the first part of the solution will go to infinity with increasing x, while in the case of a < 0 the same is true for the second part of the solution. The solution is not stable, independent of the value of b, i.e., m can be real or complex.
a = 0, i.e. m = ib, giving as the complete solution:
At this point a distinction can be made based on whether b is integer or not:
1. b integer: The solutions, called Mathieu functions of integral order, are periodic but unstable.
2. b is not integer: The solutions are periodic and stable.

Stability Diagram

These mathematical results are important, since they show us that depending on the a and q values of our system, we may have stable, or unstable solutions. A graph that shows the regions of stability of the solutions depending on the parameters in the differential equation are called stability diagrams.

Below is a part of the stability diagram for the Mathieu equation. The filled regions are regions of stable solutions, where b is not integer. The borders between the regions of stability and instability, where b is integer, are called characteristic curves.