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Geodesic Rays of the N-Body Problem

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Abstract

For the Newtonian N-body problem, we study the Jacobi–Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More precisely, we prove that the evolution of such motions asymptotically decomposes into free particles and subsystems in completely parabolic expansion. The theorem applies, in particular, to the maximal characteristic curves of any given global viscosity solution of the stationary Hamilton–Jacobi equation \(H(x,d_xu)=h\).

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Acknowledgements

The first author is grateful to Cátedras Conacyt program of Consejo Nacional de Ciencia y Tecnología del Gobierno de México. The second author thanks Grupo CSIC 618 UdelaR, Uruguay, as well as the mathematics department of Cinvestav (Instituto Politécnico Nacional, México) for the hospitality during part of the development of this work.

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Authors and Affiliations

  1. Cinvestav, Mexico City, Mexico

    J. M. Burgos

  2. IMERL, Universidad de la República, Montevideo, Uruguay

    E. Maderna

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  1. J. M. Burgos
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  2. E. Maderna
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Correspondence to E. Maderna.

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Communicated by P.-L. Lions.

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Burgos, J.M., Maderna, E. Geodesic Rays of the N-Body Problem. Arch Rational Mech Anal 243, 807–827 (2022). https://doi.org/10.1007/s00205-021-01743-3

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