The three-body problem with an inverse square law potential

Abstract

The super-separability of the three-body inverse-square Calogero systemWe study the motion of three masses in a plane interacting with a central potential proportional to 1/r 2 using the coordinates introduced recently by Piña. We show that this problem with four degrees of freedom ͑three angles and a distance related to the inertia moment of the system in these coordinates͒ is partially separable, and can be reduced to a problem with two degrees of freedom ͑two angles͒ with a new constant of motion. We find a symmetry of reflection ͑an involution͒ for this system and we use the symmetry lines to find periodic orbits in the angular coordinates. These orbits will not be periodic in general on the whole phase space because the coordinate of distance type grows as t when t→ϱ and it is unbounded. However, if the inertia moment of the system remains constant, they will be periodic on the whole phase space.

Hyperbolic Geometry for the Binary Collision Angles of the Three-Body Problem in the Plane

“…The same reference is safe for the inequalities (2.14) and (2.15). We observed at that moment that the properties of the rigid triangle are independent of the choice of the origin of σ which was selected different in other papers [1][2][3].…”

“…In references [1][2][3] however one of us presented our coordinates as if the masses were different with the definition that appears in Sect. 2 of this paper.…”

“…Equation (4.6) was deduced at reference [3] for the three relative positions of the Euler's collinear case. We prove now that this equation is also satisfied for other cases.…”

“…The masses of the three bodies are denoted by m 1 , m 2 and m 3 . In terms of the inertial coordinates x i , y i , (i = 1, 2, 3) with origin at the center of mass, our coordinates could be written as a product of three lineal transformations:…”

“…The study of this dynamics is made in the coordinate system of Piña and Jiménez [1][2][3], that is further simplified by the previous hypothesis of plane two-dimensional motion.…”