Gauge Symmetry of the N-body Problem in the Hamilton-Jacobi Approach
Abstract
In most books the Delannay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delannay system. A standard trick is then used to generalise the approach to the N-body case. We re-examine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed by "hand" to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations).
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 12, 2004
- Accession Number
- ADA423236
Entities
People
- Michael Efroimsky
- Peter Goldreich
Organizations
- United States Naval Observatory