Internal or shape coordinates in the<i>n</i>-body problem

Littlejohn, Robert G.; Reinsch, Matthias

doi:10.1103/physreva.52.2035

Cited by 80 publications

(107 citation statements)

References 18 publications

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“…We have shown that in the line configurations (corresponding to the boundaries of the reduced space) no curvature singularity occurs. The boundary conditions (3) and (4) imply that the geodesics reach the boundaries tangentially after which they must return to the inner region.…”

Section: Discussionmentioning

confidence: 99%

“…Topologically the region inside the boundaries is the quotient space R 9 /E(3), where E(3) = R 3 ⋊ SO(3) and is homeomorphic to half of R 3 . The boundary is homeomorphic to R 2 [4]. The special situation of the restricted three-body problem, with a lighter third body orbiting about the other two heavier, whose separation is not changing is confined within intersections of the space of orbits (the "beam") with one of the planes a i = const.…”

Section: The Space Of Orbitsmentioning

confidence: 99%

“…4. There we give the reduced metric on the space of relative separations first in terms of the distances and second in terms of a radial coordinate and suitably chosen angular variables.…”

Section: Introductionmentioning

confidence: 99%

“…Before proceeding with the analysis of the metric and curvature properties of the reduced configuration space, we briefly describe and ‡ 3N − 6 = N (N − 1)/2 holds also for N = 4. A discussion of the adequate coordinates for this case can be found in [4]. …”

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confidence: 99%

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

Gergely

McKain

2000

Class. Quantum Grav.

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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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Section: Discussionmentioning

confidence: 99%

Section: The Space Of Orbitsmentioning

confidence: 99%

“…4. There we give the reduced metric on the space of relative separations first in terms of the distances and second in terms of a radial coordinate and suitably chosen angular variables.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Gergely

McKain

2000

Class. Quantum Grav.

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“…This is shown by Narasimhan and Ramadas (1979) and discussed further by Littlejohn and Reinsch (1995). In the following two tetrahedra will be considered to have the same shape if they are related by a translation and a proper rotation.…”

Section: Tetrahedra and Their Shapesmentioning

confidence: 95%

Symplectic and semiclassical aspects of the Schläfli identity

Hedeman

Kur

Littlejohn

et al. 2015

J. Phys. A: Math. Theor.

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Abstract. The Schläfli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, 3-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schläfli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to 3-dimensional spaces of constant (nonzero) curvature, involving Poisson-Lie groups and q-deformed spin networks, are discussed.

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Recent Advances in the Theory of Vibration–Rotation Hamiltonians

Pesonen

Halonen

2003

Advances in Chemical Physics

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Internal or shape coordinates in then-body problem

Cited by 80 publications

References 18 publications

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

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

Symplectic and semiclassical aspects of the Schläfli identity

Recent Advances in the Theory of Vibration–Rotation Hamiltonians

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