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Asymptotic velocity for four celestial bodies.

Andreas Knauf1

  • 1Department of Mathematics, Friedrich-Alexander-University Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany knauf@math.fau.de.

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Asymptotic velocities, the Cesàro limit of velocity, are shown to exist for systems with up to four bodies. This finding applies to specific pair potentials in higher dimensions and nearly all initial conditions.

Keywords:
asymptotic velocitycelestial mechanicsscattering theory

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Area of Science:

  • Celestial mechanics
  • Dynamical systems theory
  • Mathematical physics

Background:

  • Asymptotic velocity, defined as the Cesàro limit of velocity, has been proven to exist for bounded interaction potentials.
  • Its existence is known to be incorrect in celestial mechanics involving four or more bodies.
  • This study addresses the conditions under which asymptotic velocities can exist in multi-body systems.

Purpose of the Study:

  • To investigate the existence of asymptotic velocities for a class of pair potentials in multi-body systems.
  • To determine the conditions (number of bodies, dimensions, energy, initial conditions) for the existence of asymptotic velocities.
  • To extend the understanding of asymptotic velocity beyond bounded potentials.

Main Methods:

  • Analysis of pair potentials, specifically homogeneous potentials of degree -α for α∈(0, 2).
  • Consideration of systems with up to four bodies in three or more dimensions.
  • Examination of the energy surface and initial conditions.

Main Results:

  • Asymptotic velocities are shown to exist for systems with up to four bodies in three or more dimensions.
  • Existence is demonstrated for a class of pair potentials, including homogeneous ones.
  • The result holds for any energy and almost all initial conditions on the energy surface.

Conclusions:

  • The existence of asymptotic velocities is confirmed for a broader range of conditions than previously established.
  • This work contributes to the understanding of long-term dynamics in multi-body systems.
  • Findings are relevant to the theme issue on 'Finite dimensional integrable systems: new trends and methods'.