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M. Efroimsky (US Naval Observatory), P. Goldreich (CalTech)
In most books the Lagrange and Delaunay systems of equations for the orbital elements are derived in the Hamilton-Jacobi approach: one begins with two-body Hamilton equations in spherical or Cartesian coordinates; then carries out a canonical transformation to the orbital elements and, thus, arrives to the Delaunay or Lagrange system. A standard trick then enables one to generalize the approach to the N-body case. We carefully re-examine this step and demonstrate that it contains an implicit condition which restricts the orbit to a certain 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. This tacit assumption is equivalent to the so-called Lagrange constraint, one that Lagrange imposed ``by hand'' in order to remove the excessive freedom, when he was deriving his system of equations by the method of variation of parameters.
The physical meaning of this implicit condition, tacitly present also in the Hamilton-Jacobi treatment of the N-body problem, is transparent: it is the condition of the orbital elements being osculating (i.e., of the velocity being expressed through the orbital elements in the same manner as in the two-body case). The imposure of any supplementary condition, which is different from the Lagrange constraint (but is compatible with the equations of motion), is legitimate. However, it will alter the form of the Lagrange and Delaunay equations (Efroimsky 2002, Newman & Efroimsky 2003) and will have consequences for numerical integrators (Efroimsky 2002, Murison & Efroimsky 2003).
Another important alteration of the Lagrange and Delaunay systems will be in order when the disturbing function depends not only upon the coordinates but also upon the velocities, i.e., when the orbital elements are defined in a non-inertial coordinate system (Goldreich 1965).
In the current presentation we consider interplay between these two issues: the freedom of gauge fixing and the freedom of reference-system choice. We apply our results to description of a satellite motion about a precessing planet.
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Bulletin of the American Astronomical Society, 35 #4
© 2003. The American Astronomical Soceity.