Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature

Shchepetilov, A. V.

doi:10.1088/0305-4470/31/29/017

Cited by 33 publications

(27 citation statements)

References 9 publications

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“…Translational invariance guarantees a conserved total momentum, but since there are no boost symmetries, this cannot be split off as a separate centre of mass motion. In general if one considers two particles in H 3 interacting by a force which depends only on their separation, one finds that the relative motion depends on the total momentum of the system [23], [24]. Thus, it is not unreasonable to expect that the moduli space metric will not split up in this fashion.…”

Section: Hyperbolic Monopolesmentioning

confidence: 99%

Hidden symmetry of hyperbolic monopole motion

Gibbons

Warnick

2007

Journal of Geometry and Physics

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Hyperbolic monopole motion is studied for well separated monopoles. It is shown that the motion of a hyperbolic monopole in the presence of one or more fixed monopoles is equivalent to geodesic motion on a particular submanifold of the full moduli space. The metric on this submanifold is found to be a generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The one centre case is analysed in detail as a special case of a class of systems admitting a conserved Runge-Lenz vector. The two centre problem is also considered. An integrable classical string motion is exhibited.

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Section: Hyperbolic Monopolesmentioning

confidence: 99%

Hidden symmetry of hyperbolic monopole motion

Gibbons

Warnick

2007

Journal of Geometry and Physics

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“…Several members of the Russian school of celestial mechanics, including Valeri Kozlov and Alexander Harin, [43], [45], Alexey Borisov, Ivan Mamaev, and Alexander Kilin, [5], [6], [7], [8], [39], Alexey Shchepetilov, [67], [68], [69], and Tatiana Vozmischeva, [74], revisited the idea of the cotangent potential for the 2-body problem and considered related problems in spaces of constant curvature starting with the 1990s. The main reason for which Kozlov and Harin supported this approach was mathematical.…”

Section: Introductionmentioning

confidence: 99%

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

2012

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Abstract. We generalize the Newtonian n-body problem to spaces of curvature κ = constant, and study the motion in the 2-dimensional case. For κ > 0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for κ < 0. In the end, we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. We also emphasize that fixed points are specific to the case κ > 0, hyperbolic relative equilibria to κ < 0, and Lagrangian orbits of arbitrary masses to κ = 0-results that provide new criteria towards understanding the large-scale geometry of the physical space.

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“…We should also mention that there have been recent efforts in formulating mechanics on non-Euclidean spaces, e.g. [12,13]. Suppose the interacting particles lie in a Riemannnian manifold (S, g), which we assume is geodesically complete.…”

Section: Energy Balance For Particle Systems On Riemannian Manifoldsmentioning

confidence: 99%

Energy balance invariance for interacting particle systems

Yavari

Marsden

2008

Z. Angew. Math. Phys.

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This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However, the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws.These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics at their end points. One also obtains a discrete version of the Doyle-Ericksen formula, which relates the magnitude of internal forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background metric.

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Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature

Cited by 33 publications

References 9 publications

Hidden symmetry of hyperbolic monopole motion

Hidden symmetry of hyperbolic monopole motion

The n-Body Problem in Spaces of Constant Curvature. Part II: Singularities

Energy balance invariance for interacting particle systems

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