Saari's Homographic Conjecture of the Three-Body Problem

Abstract

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.

References in corpus (4)

• Euler configurations and quasi-polynomial systems
• Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum
• A Counterexample to a Generalized Saari's Conjecture with a Continuum of Central Configurations
• Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography

Cited by in corpus (7)

• An intrinsic approach in the curved n-body problem: the negative curvature case
• Relative equilibria in the 3-dimensional curved n-body problem
• On the stability of tetrahedral relative equilibria in the positively curved 4-body problem
• Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem
• Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential
• Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity
• Saari's homographic conjecture for planar equal-mass three-body problem under a strong force potential