Three point masses attract one another by Newtonian gravity. Newton posed the question in the 1680s; in 1887 Bruns showed there are no algebraic first integrals beyond energy, momentum and angular momentum, and Poincaré's prize memoir of 1890 showed why: generic three-body motion is chaotic. So there is no general solution in elementary functions, and the "known solutions" are special motions, almost all of them periodic.
Exact solutions
Euler (1767) and Lagrange (1772) found the only motions in which the triangle keeps its shape: the collinear and the equilateral central configurations. For any three masses there are exactly five of them up to rotation and scaling — three collinear orderings and the equilateral triangle in its two orientations — and each can move on circles or on similar Kepler ellipses of any eccentricity. These are the two "Central configurations" entries; their masses and eccentricity are adjustable.
Sundman (1912) proved that the planar problem with nonzero angular momentum has a convergent power-series solution valid for all time. It is a genuine general solution, but it converges so slowly that it is useless for computation, and it is not shown here.
Periodic orbits found by computation
Broucke, Hénon and Hadjidemetriou (1975–76) mapped families of planar periodic orbits for equal masses; six members are included. Moore (1993) found the figure-eight, in which three equal masses chase one another along a single curve; Chenciner and Montgomery (2000) proved it exists, and it is linearly stable. Šuvakov and Dmitrašinović (2013) searched the zero-angular-momentum, equal-mass case and reported 13 new orbits in 11 families; they are all here under their published names. Since then Li and Liao have pushed the count into the thousands: 695 equal-mass families in 2017, 1,349 unequal-mass families in 2018, about 600,000 Broucke–Hadjidemetriou–Hénon orbits and satellites in 2020, and 10,059 three-dimensional periodic orbits in 2025, including 21 spatial choreographies. Those catalogues are far too large to embed; the Orbit finder searches the same isosceles-collinear initial-condition plane and will rediscover members of them on your machine.
The restricted problem
When the third mass is negligible the problem is the circular restricted three-body problem. It has the five Lagrange points and rich families of periodic orbits around them: planar Lyapunov orbits, the three-dimensional halo orbits proposed by Farquhar (1968) and first flown by ISEE-3, distant retrograde orbits, and Arenstorf's 1963 Earth–Moon figure. Trojan tadpoles and horseshoes are quasi-periodic librations around L4/L5 and L3. Sitnikov's problem (1960) puts a massless body on the axis of an eccentric binary; Sitnikov used it to prove that oscillatory motions exist, and it became a standard model of chaotic motion.
What this page computes
Every entry is integrated live with an adaptive Dormand–Prince 5(4) method with G = 1. Initial conditions for the periodic orbits were refined here by Newton shooting at tolerance 10⁻¹³. The closure figure in the side panel, |y(T) − y(0)| in phase space, is recomputed on every run: at the default tolerance it is typically 10⁻⁹–10⁻⁸, while Butterfly IV (about 10⁻⁴) and the yin-yang II pair (about 10⁻⁶) are so unstable that double precision cannot do much better. Use Perturb and Periods to see which orbits survive a small kick. The shape sphere shows the triangle's shape independent of size and orientation: the poles are the two Lagrange triangles and the three marked equator points are binary collisions.
Not included: Schubart's 1956 collinear orbit and other orbits that pass through binary collisions, which need regularised coordinates rather than a plain integrator. The Pythagorean problem (masses 3, 4, 5) is included as the classic non-periodic example; it has extremely close encounters, so its final escape direction is robust but the exact ejection time depends on the tolerance.
References
Euler, Novi Comm. Acad. Petrop. 11 (1767). Lagrange, Essai sur le problème des trois corps (1772). Burrau, Astron. Nachr. 195 (1913). Sitnikov, Dokl. Akad. Nauk 133 (1960). Arenstorf, Am. J. Math. 85 (1963). Szebehely & Peters, Astron. J. 72 (1967). Broucke, Celest. Mech. 12 (1975). Hénon, Celest. Mech. 13 (1976). Moore, Phys. Rev. Lett. 70 (1993). Chenciner & Montgomery, Ann. Math. 152 (2000). Šuvakov & Dmitrašinović, Phys. Rev. Lett. 110, 114301 (2013). Li & Liao, Sci. China PMA 60 (2017), arXiv:1705.00527. Li, Jing & Liao, PASJ 70 (2018), arXiv:1709.04775. Li & Liao, arXiv:2008.13550 (2020). Li & Liao, arXiv:2508.08568 (2025).