**34th Meeting of the AAS Division on Dynamical Astronomy, May 2003**

* 9 Standards and Gauges*

Oral, Tuesday, May 6, 2003, 1:00-3:05pm,
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## [9.02] The Method of Variation of Constants and Multiple Time Scales in Orbital Mechanics

*W.I. Newman (University of California), M. Efroimsky (U.S. Naval Observatory)*

The method of variation of constants is an important tool
used to solve systems of ordinary differential equations,
and was invented by Euler and Lagrange to solve a problem in
orbital mechanics. This methodology assumes that certain
``constants'' associated with a homogeneous problem will
vary in time in response to an external force. It also
introduces one or more constraint equations motivated by the
nature of the time-dependent driver. We show that these
constraints can be generalized, in analogy to gauge theories
in physics, and that different constraints can offer
conceptual advances and methodological benefits to the
solution of the underlying problem. Examples are given from
linear ordinary differential equation theory and from
orbital mechanics. However, a slow driving force in the
presence of multiple time scales contained in the underlying
(homogeneous) problem nevertheless requires special care,
and this has strong implications to the analytic and
numerical solutions of problems ranging from celestial
mechanics to molecular dynamics.

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Bulletin of the American Astronomical Society, **35** #4

© 2003. The American Astronomical Soceity.