Given a system of periodic Ordinary Differential Equations of the form

(1) |

(2) |

(3) |

(4) | |||

(5) | |||

(6) |

Plugging these into (3) gives

(7) |

(8) |

(9) |

(10) |

Integrating gives

(11) |

(12) |

(13) |

(14) |

and

(15) |

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

(16) |

**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.

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1999-05-26