Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum

Abstract

Geometrical properties of three-body orbits with zero angular momentum are investigated. If the moment of inertia is also constant along the orbit, the triangle whose vertexes are the positions of the bodies, and the triangle whose perimeters are the momenta of the bodies, are always similar (``synchronised similar triangles''). This similarity yields kinematic equalities between mutual distances and magnitude of momenta. Moreover, if the orbit is a solution to the equation of motion under homogeneous potential, the orbit has a new constant involving momenta. For orbits with zero angular momentum and non-constant moment of inertia, we introduce scaled variables, positions divided by square root of the moment of inertia and momenta derived from the velocity of the scaled positions. Then the similarity and the kinematic equalities hold for the scaled variables. Using this similarity, we prove that any bounded three-body orbit with zero angular momentum under homogeneous potential whose degree is smaller than 2 has infinitely many collinear configurations (syzygies or eclipses) or collisions.

14 pages, 3 figures, submitted to The Journal of Physics A

References in corpus (3)

• An existence of simple choreographies for N-body problem - a computer assisted proof
• Choreographic Three Bodies on the Lemniscate
• Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography

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• Computable Integrability. Chapter 1: General notions and ideas
• Global geometry of 3-body motions with vanishing angular momentum, I
• Fitting Hyperbolic Pants to a Three-Body Problem