Given a system of ordinary differential equations of the form

(1)

that are periodic in , the solution can be written as a linear combination of functions of the form

(2)

where  is a function periodic with the same period  as the equations themselves. Given an ordinary differential equation of the form

(3)

where  is periodic with period , the ODE has a pair of independent solutions given by the real and imaginary parts of

			(4)
			(5)
			(6)
			(7)

Plugging these into (◇) gives

(8)

so the real and imaginary parts are

(9)

(10)

From (◇),

			(11)
			(12)
			(13)

Integrating gives

(14)

where  is a constant which must equal 1, so  is given by

(15)

The real solution is then

(16)

so

			(17)
			(18)
			(19)
			(20)

and

			(21)
			(22)
			(23)
			(24)

which is an integral of motion. Therefore, although  is not explicitly known, an integral  always exists. Plugging (◇) into (◇) gives

(25)

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

REFERENCES:

Abramowitz, M. and Stegun, I. 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.