Deprit’s reduction of the nodes revisited

Abstract

We revisit a set of symplectic variables introduced by Andre Deprit in [5], which allows for a complete symplectic reduction in rotation invariant Hamiltonian systems, generalizing to arbitrary dimension Jacobi's reduction of the nodes. In particular, we introduce an action-angle version of Deprit's variables, connected to the Delaunay variables, and give a new hierarchical proof of the symplectic character of Deprit's variables.

Flow reconstruction and invariant tori in the spatial three-body problem

“…For a detailed description of these variables the reader can consult [7] and also [39]. For k = 1, 2, half of Deprit's coordinates, namely k , L k , G k , coincide with the spatial Delaunay variables, specifically for the ellipse k, k designates the mean anomaly, L k stands for its conjugate momentum, i.e.…”

“…These variables were designed in [17] for eliminating two nodal angles in the N -body problem. Deprit's elements were used by Ferrer and Osácar in their study of the three-body problem [23] and by Chierchia and Pinzari to determine invariant tori of the spatial N -body problem in [6][7][8][9]. The rotational reduction is singular in the sense of Arms, Cushman and Gotay [2], see also [12][13][14].…”

Partial reduction and Delaunay/Deprit variables

“…the ellipse they describe are non-degenerate, non-circular, non-horizontal. 4 The terminology follows from [6]. •ḡ 1 ,ḡ 2 denote the angles from ν L 6 to the pericentres;…”

“…Nevertheless, their sum φ 1 + φ 2 remains well defined. One can then recover Jacobi's elimination of the node from the Deprit variables by a limit procedure, see [6] for details. More explicit and precise definitions of these variables can be found in [6].…”

“…One can then recover Jacobi's elimination of the node from the Deprit variables by a limit procedure, see [6] for details. More explicit and precise definitions of these variables can be found in [6]. As mentioned above, the symplecticity of these set of coordinates is proven by Chierchia and Pinzari [6] (Deprit proved the symplecticity of his set of coordinates in [7]).…”