The planetary N-body problem: symplectic foliation, reductions and invariant tori

Deprit’s reduction of the nodes revisited

“…But in contrast with Jacobi's celebrated result, Deprit's variables seem to be not well known 1 and often believed to be unpractical (as mentioned by Deprit himself [5, p. 194] or, e.g., in 2 [7]). On the contrary, Deprit's variable may be very useful and can be effectively used (after a suitable "Poincaré regularization") to compute, for example, Birkhoff normal forms for the planetary (1 + n)-body problem or to check KAM nondegeneracies; compare [4], [3]. Our presentation differs from that of Deprit in two respects.…”

“…In fact is it not difficult (see [4,Appendix A.1]) to write down inverse formulae showing that the map…”

“…with the (Ψ, ψ)'s as in (4). He proved that such variables are homogeneoussymplectic with respect to the Cartesian variables (y, x), namely, that the 1-form y · dx is preserved by the transformation (R, Φ, Ψ, r, ϕ, ψ) → (y, x).…”

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

“…Lochak and Porzio, 1989) amount to a substantial research project in their own right. It is worth mentioning that it is only in the last 2 years that the nondegeneracy condition for the KAM theorem in the context of the general n body problem of celestial mechanics has been verified (the work of Chierchia and Pinzari, 2011). Nevertheless, even though its rigorous applicability to the n body problem was not established, the KAM theorem provided a valuable theoretical framework for thinking about the problem.…”

Partial reduction and Delaunay/Deprit variables

“…Note that in both of these references, Deprit coordinates are built for the general N-body problem, with the aim to generalize Jacobi's elimination of nodes, or to conveniently reduce the SO(3)-symmetry of the N-body problem for N ≥ 4, which is of significant importance for the perturbative study of the N-body problem (c.f. [8]). …”