A remarkable periodic solution of the three-body problem in the case of equal masses

By Alain Chenciner, Richard Montgomery. 2000

608 citations recorded by the source corpus.

Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a flxed eight-shaped curve. Setting aside collinear motions, the only other known motion along a flxed curve in the inertial plane is the \Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every \Euler conflguration" in which one of the bodies sits at the midpoint of the segment deflned by the other two (Figure 1). Numerical computations

Subfields: Aerospace Engineering, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/a4f222bee95204bd93078cc8e558b5a8d04c299d

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Citations in the corpus

Continuation of periodic orbits in conservative and Hamiltonian systems (2003, methodology): “An explicit example of a Hamiltonian system with a scaling property is given by the n-body problem; in particular, all so-called figure-eight orbits of the 3-body problem with equal masses (see [3]) are related by scaling, and starting from one such orbit our continuation method will give the full b”
Timescales of Kozai-Lidov oscillations at quadrupole and octupole order in the test particle limit (2015, background): “…of supermassive and intermediatemass black holes (Miller & Hamilton 2002b; Blaes et al. 1 While stable non-hierarchical triple systems are possible (e.g., Chenciner & Montgomery 2000; Šuvakov & Dmitrašinović 2013), they may require fine tuning to form and have never been observed in nature.”
How the Method of Minimization of Action Avoids Singularities (2002, methodology): “The remarkable periodic solution presented in [ 1 ] uses a proof of existence based on the method of minimization of action, and the two authors have been very careful about the necessity to verify the absence of singularities.”
Stellar collisions during binary–binary and binary–single star interactions (2004, background): “It should also be noted that our use of binary trees prevents us from recognizing stable three-body systems that are not hierarchical, such as the stable ‘figure eight’ orbit for three stars of comparable mass (Chenciner & Montgomery 2000; Montgomery 2001).”
Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit (2009, methodology): “r12 = r13) which are used in [ CM ]t o construct the eight.”
New Families of Solutions in N -Body Problems (2001, methodology): “The proof of the existence (see [4]) involves a variational argument.”
Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry (2003, background): “ 3−u) 2 3 of the horizontal relative equilibrium solution xu of an equilateral triangle which rotates by an angle (π 3 −u) during the given amount of time. The ﬁrst estimation, better than the one in [CM] (Aˆ 2 = 2 1 3A2) , appears in [ZZ2] in the case of the Eight. It follows from the remark (at th”
Action-Minimizing Orbits in the Parallelogram Four-Body Problem with Equal Masses (2001, methodology): “This lemma follows by exactly the same arguments as in [5].”
On the existence of collisionless equivariant minimizers for the classical n-body problem (2003, background): “…system s:n-body problem s);Secondary 70F16 (M echanics ofparticles and system s: Collisions in celestialm echanics,regularization),37C80 (Dynam icalsystem sand ergodic theory: Sym m etries, equivariant dynam icalsystem s), 70G 75 (M echanics of particlesand system s:Variationalm ethods).”
Gravitational N-Body Simulations (2003, background): “Another example was recently revealed by the existence proof for a periodic 2D orbit where three equal masses describe a figure eight solution [Chenciner & Montgomery, 2000].”