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

TL;DR

This paper explores geometric properties of three-body orbits with zero angular momentum, revealing conditions for similarity of triangles formed by positions and momenta, and proving the existence of infinitely many collinear configurations under certain potentials.

Contribution

It introduces the concept of synchronised similar triangles in three-body orbits and establishes new geometric and dynamical invariants for such systems.

Findings

Triangles of positions and momenta are similar in specific conditions.

A new constant involving momenta is identified for certain orbits.

Bounded orbits under homogeneous potentials with degree less than 2 have infinitely many collinear configurations.

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.