TY - JOUR

T1 - Natural dynamical reduction of the three-body problem

AU - Kol, Barak

N1 - Funding Information: I thank Yogesh Dandekar, Lior Lederer and Subhajit Mazumdar for collaboration on a related project. This research, as well as Kol (), was performed during the times of the COVID-19 pandemic and benefitted from the associated isolation. Part of this research was supported by the Israel Science Foundation (Grant No. 1345/21). I dedicate this work to the memory of Ami Nathan, 15.6.1934-24.10.2020, my father in law and a dedicated family man. Publisher Copyright: © 2023, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2023/6

Y1 - 2023/6

N2 - The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation of a mechanical problem to fewer degrees of freedom, a process known as dynamical reduction. However, extant reductions are either non-general, or hide the problem’s symmetry or include unexplained definitions. This paper presents a general and natural dynamical reduction, which avoids these issues. Any three-body configuration defines a triangle, and its orientation in space. Accordingly, we decompose the dynamical variables into the geometry (shape + size) and orientation of the triangle. The geometry variables are shown to describe the motion of an abstract point in a curved 3d space, subject to a potential-derived force and a magnetic-like force with a monopole charge. The orientation variables are shown to obey a dynamics analogous to the Euler equations for a rotating rigid body; only here the moments of inertia depend on the geometry variables, rather than being constant. The reduction rests on a novel symmetric solution to the center of mass constraint inspired by Lagrange’s solution to the cubic. The formulation of the orientation variables is novel and rests on a partially known generalization of the Euler–Lagrange equations to non-coordinate velocities. Applications to global features, to the statistical solution, to special exact solutions and to economized simulations are presented. A generalization to the four-body problem is presented.

AB - The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation of a mechanical problem to fewer degrees of freedom, a process known as dynamical reduction. However, extant reductions are either non-general, or hide the problem’s symmetry or include unexplained definitions. This paper presents a general and natural dynamical reduction, which avoids these issues. Any three-body configuration defines a triangle, and its orientation in space. Accordingly, we decompose the dynamical variables into the geometry (shape + size) and orientation of the triangle. The geometry variables are shown to describe the motion of an abstract point in a curved 3d space, subject to a potential-derived force and a magnetic-like force with a monopole charge. The orientation variables are shown to obey a dynamics analogous to the Euler equations for a rotating rigid body; only here the moments of inertia depend on the geometry variables, rather than being constant. The reduction rests on a novel symmetric solution to the center of mass constraint inspired by Lagrange’s solution to the cubic. The formulation of the orientation variables is novel and rests on a partially known generalization of the Euler–Lagrange equations to non-coordinate velocities. Applications to global features, to the statistical solution, to special exact solutions and to economized simulations are presented. A generalization to the four-body problem is presented.

UR - http://www.scopus.com/inward/record.url?scp=85160631267&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10569-023-10144-5

DO - https://doi.org/10.1007/s10569-023-10144-5

M3 - Article

C2 - 37215293

SN - 0923-2958

VL - 135

JO - Celestial Mechanics and Dynamical Astronomy

JF - Celestial Mechanics and Dynamical Astronomy

IS - 3

M1 - 29

ER -