Abstract
The partial reduction procedure of the rotational symmetry of the N-body problem means to only fix the direction of the total angular momentum, leaving the norm of the total angular momentum unfixed and the rotational symmetry around this direction unreduced. In this article, we present a conceptual link between this procedure, appearing as an intermediate step in the total reduction procedure of the rotational symmetry, with the symplectic cross-section theorem of Guillemin–Sternberg. As an application of this link, we present some alternative proofs of the symplecticity of the Delaunay and Deprit coordinates, which are important symplectic coordinates in the perturbative study of celestial mechanics.
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Notes
Fulton and Harris (1991) provides a nice presentation of all the involved notions in the theory of Lie groups and Lie algebras.
We have identified \(\mathfrak {so}^{*}(3)\), the space of \(3 \times 3\) anti-symmetric matrices, with \(\mathbb {R}^{3}\) in the standard way.
I.e. the horizontal plane.
As is such called in Chierchia and Pinzari (2011a).
This is the common node line of the two planes in the Laplace plane.
A conventional choice of orientation of the node line, is given by their ascending nodes, which leads to opposite orientations of \(\nu _{L}\) in the definition of \(\bar{g}_{1}\) and \(\bar{g}_{2}\).
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Acknowledgments
The result presented in Sect. 4 of this article are derived from my thesis Zhao (2013) prepared at the Paris Observatory and the Paris Diderot University. Many thanks to Alain Chenciner and Jacques Féjoz, my thesis supervisors, for their constant help during years, and to them and to Henk Broer, for comments and suggestions.
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Zhao, L. Partial reduction and Delaunay/Deprit variables. Celest Mech Dyn Astr 120, 423–432 (2014). https://doi.org/10.1007/s10569-014-9584-1
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DOI: https://doi.org/10.1007/s10569-014-9584-1