The geometry of the Barbour-Bertotti theories: II. The three-body problem

Abstract

Abstract. We present a geometric approach to the three-body problem in the non-relativistic context of the Barbour-Bertotti theories. The Riemannian metric characterizing the dynamics is analyzed in detail in terms of the relative separations. Consequences of a conformal symmetry are exploited and the sectional curvatures of geometrically preferred surfaces are computed. The geodesic motions are integrated. Line configurations, which lead to curvature singularities for N = 3, are investigated. None of the independent scalars formed from the metric and curvature tensor diverges there. IntroductionThe nonrelativistic dynamical models of Barbour and Bertotti [1,2] arose from the criticism of the concepts of absolute space and time. They describe a classical interacting N −particle system subjected to the Hamiltonian, momenta and angular momenta constraints. The invariance group of the theory is the Leibniz group [1], which includes time-dependent translations and rotations together with the monotonous but otherwise arbitrary redefinition of time.In a previous paper [3] being referred hereafter as paper I, one of the present authors has analyzed the underlying geometry of the Barbour-Bertotti theories. The reduction process on the Lagrangian was carried out by solving the constraints, arriving to a Riemannian line element.This reduction was possible for all configurations but the line ones. The emerging Riemannian metric G was shown to represent the first fundamental form of the orbit space of the Leibniz group. The geodesics in this metric characterize the free motions, pertinent to constant potential V . For a generic potential V = const the motions are geodesics of the conformally scaled Jacobi metric −2V (x)G.In I the Riemann tensor and curvature scalar were computed in terms of the vorticity tensors of the generators of rotations. Then the curvature scalar was expressed in terms of the principal moments of inertia and the number of particles. An analysis based on this expression allowed us to conclude that the line configurations represent curvature singularities for N = 3. One would like to say more about these configurations for the exceptional case N = 3. This is one of the motivations of the present work. 2The second motivation for specializing to three particles is the remarkable coincidence between the number of relative separations N (N −1)/2 among the particles and the dimension 3N − 6 of the reduced space ‡ . This feature enables one to employ the distances as a symmetric set of variables of the space of orbits and to analyze in detail the underlying Riemannian geometry.We discuss and picture the space of orbits in Sec. 2. Similar discussions can be found in [5] and [6]. The space of orbits being a manifold with boundary, we announce and prove the conditions a geodesic reaching this boundary has to obey.The rest of the paper is organized as follows. First we particularize the generic expression of the curvature scalar obtained in I to the case of three particles. For this purpose we compute in Sec. 3 t...

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