Natural dynamical reduction of the three-body problem. (arXiv:2107.12372v1 [astro-ph.EP])
<a href="http://arxiv.org/find/astro-ph/1/au:+Kol_B/0/1/0/all/0/1">Barak Kol</a>
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 dynamical reduction that avoids these issues, and hence
is general and natural. 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 little known
generalization of the Euler-Lagrange equations to non-coordinate velocities.
Applications to special exact solutions and to the statistical solution are
described or discussed. Moreover, a generalization to the four-body problem is
presented.
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 dynamical reduction that avoids these issues, and hence
is general and natural. 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 little known
generalization of the Euler-Lagrange equations to non-coordinate velocities.
Applications to special exact solutions and to the statistical solution are
described or discussed. Moreover, a generalization to the four-body problem is
presented.
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