Natural dynamical reduction of the three-body problem.

Area of Science:

• Physics
• Astrophysics
• Celestial Mechanics

Background:

• The three-body problem is a fundamental challenge in physics with wide-ranging applications.
• Existing dynamical reductions often lack generality, obscure symmetries, or use unexplained definitions.

Purpose of the Study:

• To present a general and natural dynamical reduction for the three-body problem.
• To overcome limitations of extant reduction methods.

Main Methods:

• Decomposing dynamical variables into the geometry (shape, size) and orientation of the three-body configuration triangle.
• Utilizing a novel symmetric solution to the center of mass constraint.
• Applying a generalization of Euler-Lagrange equations to non-coordinate velocities.

Main Results:

• Geometry variables describe motion in a curved 3D space with potential and magnetic-like forces.
• Orientation variables follow dynamics analogous to Euler's rigid body equations, with geometry-dependent moments of inertia.
• The reduction is applied to global features, statistical solutions, exact solutions, and simulations.

Conclusions:

• The presented dynamical reduction offers a more general and natural approach to the three-body problem.
• This formulation simplifies analysis and simulation, with potential extensions to the four-body problem.