Abstract
The aim of this paper is to study the global geometry of non-planar 3-body motions in the realms of equivariant Differential Geometry and Geometric Mechanics. This work was intended as an attempt at bringing together these two areas, in which geometric methods play the major role, in the study of the 3-body problem. It is shown that the Euler equations of a three-body system with non-planar motion introduce non-holonomic constraints into the Lagrangian formulation of mechanics. Applying the method of undetermined Lagrange multipliers to study the dynamics of three-body motions reduced to the level of moduli space \({\bar{M}}\) subject to the non-holonomic constraints yields the generalized Euler-Lagrange equations of non-planar three-body motions in \({\bar{M}}\) . As an application of the derived dynamical equations in the level of \({\bar{M}}\) , we completely settle the question posed by A. Wintner in his book [The analytical foundations of Celestial Mechanics, Sections 394–396, 435 and 436. Princeton University Press (1941)] on classifying the constant inclination solutions of the three-body problem.
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Salehani, M.K. Global geometry of non-planar 3-body motions. Celest Mech Dyn Astr 111, 465–479 (2011). https://doi.org/10.1007/s10569-011-9381-z
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DOI: https://doi.org/10.1007/s10569-011-9381-z