Floquet Analysis

Given a system of periodic Ordinary Differential Equations of the form

$${d\over dt} \left[{\matrix{x\cr y\cr v_x\cr v_y\cr}}\right] ... ... 0\cr}}\right]\left[{\matrix{x\cr y\cr v_x\cr v_y\cr}}\right],$$ (1)

the solution can be written as a linear combination of functions of the form

$$\left[{\matrix{x(t)\cr y(t)\cr v_x(t)\cr v_y(t)\cr}}\right] ... ...x{x_0\cr y_0\cr v_{x0}\cr v_{y0}\cr}}\right]e^{\mu t}P_\mu(t),$$ (2)

where $P_\mu(t)$ is a function periodic with the same period $T$ as the equations themselves. Given an Ordinary Differential Equation of the form

$$\ddot x+g(t)x=0,$$ (3)

where $g(t)$ is periodic with period $T$, the ODE has a pair of independent solutions given by the Real and Imaginary Parts of

$$x(t) = w(t)e^{i\psi(t)}$$ (4)

$$\dot x = (\dot w+iw\dot\psi)e^{i\psi}$$ (5)

$$\ddot x = [\ddot w+i\dot w\dot \psi+i(\dot w\dot \psi+w\ddot \psi+iw\dot\psi ^2)]e^{i\psi}$$

$$= [(\ddot w-w\dot\psi^2)+i(2\dot w\dot \psi+w\ddot \psi)]e^{i\psi}.$$ (6)

Plugging these into (3) gives

$$\ddot w+2i\dot w\dot\psi+w(g+i\ddot\psi-\dot\psi^2)=0,$$ (7)

so the Real and Imaginary Parts are

$$\ddot w+w(g-\dot\psi^2)=0$$ (8)

$$2\dot w\dot\psi+w\ddot\psi=0.$$ (9)

From (9),

$${2\dot w\over w}+{\ddot\psi\over \dot\psi} = 2 {d\over dt} (\ln w)+{d\over dt} [\ln(\dot\psi)]$$

$$= {d\over dt} \ln(\dot\psi w^2) =0.$$ (10)

Integrating gives

$$\dot\psi={C\over w^2},$$ (11)

where $C$ is a constant which must equal 1, so $\psi$ is given by

$$\psi=\int_{t_0}^t {dt\over w^2}.$$ (12)

The Real solution is then

$$x(t)=w(t)\cos[\psi(t)],$$ (13)

so

$$\dot x = \dot w\cos \psi-w\dot\psi\sin \psi = \dot w {x\over w} -w\dot\psi \sin \psi$$

$$= \dot w{x\over w}-w {1\over w^2}\sin \psi = \dot w {x\over w}-{1\over w}\sin \psi$$ (14)

and

$$1 = \cos ^2\psi+\sin ^2\psi=x^2w^{-2}+\left[{w\left({\dot w {x\over w}-\dot x}\right)}\right]^2$$

$$= x^2w^{-2}+(\dot w x-w\dot x)^2 \equiv I(x,\dot x,t),$$ (15)

which is an integral of motion. Therefore, although $w(t)$ is not explicitly known, an integral $I$ always exists. Plugging (10) into (8) gives

$$\ddot w+g(t)w-{1\over w^3}=0,$$ (16)

which, however, is not any easier to solve than (3).

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 727, 1972.

Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 175, 1987.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 32, 1983.

Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry, 2 vols. Princeton, NJ: Van Nostrand, 1956-64.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.