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Scattering Parabolic Solutions for the Spatial N-Centre Problem

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Abstract

For the N-centre problem in the three dimensional space,

$${\ddot{x}} = -\sum_{i=1}^{N} \frac{m_i \,(x-c_i)}{\vert x - c_i \vert^{\alpha+2}}, \qquad x \in {\mathbb{R}}^3 {\setminus} \{c_1,\ldots,c_N\},$$

where \({N \geqq 2}\), \({m_i > 0}\) and \({\alpha \in [1,2)}\), we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min–max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions.

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Authors and Affiliations

  1. Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto, 10, 10123, Torino, Italy

    Alberto Boscaggin, Walter Dambrosio & Susanna Terracini

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  1. Alberto Boscaggin
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  2. Walter Dambrosio
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  3. Susanna Terracini
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Correspondence to Susanna Terracini.

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Communicated by P. Rabinowitz

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Boscaggin, A., Dambrosio, W. & Terracini, S. Scattering Parabolic Solutions for the Spatial N-Centre Problem. Arch Rational Mech Anal 223, 1269–1306 (2017). https://doi.org/10.1007/s00205-016-1057-0

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