For two bodies under mutual gravity the orbits close and Kepler's laws hold exactly. Add a third body and the math falls apart — there's no closed-form solution. But if one of the three is much smaller than the others, and the two big ones move on circular orbits, a strange and beautiful structure appears: five points where a small object can sit (almost) still, indefinitely.
The full three-body problem — three masses pulling on each other, no shortcuts — has no analytic solution. Henri Poincaré proved this in the 1890s and in the process invented the field that became known as chaos theory. Any closed-form expression for where the bodies will be at time $t$ is impossible. Numerical integration is the only general way out.
But there's a special case that's tractable. The circular restricted three-body problem (CRTBP) makes three simplifying assumptions:
This describes a lot of real systems quite well: a satellite near the Sun-Earth pair (Earth's mass is tiny next to the Sun and we're ignoring the satellite's), a probe in the Earth-Moon system, an asteroid sharing Jupiter's orbit. The Earth's actual orbit isn't perfectly circular and Jupiter exists, but as a first approximation the CRTBP is excellent.
The two big bodies are perpetually moving, so describing their pull on the third body in a normal inertial frame is messy. Switching to a rotating frame that turns with the two-body system makes both primaries stand still — at the cost of two extra "fictitious" forces that any rotating frame brings along: centrifugal (outward, depending only on position) and Coriolis (perpendicular to motion, only acts when the body is moving).
Working in normalized units where the masses sum to 1, the distance between the two primaries is 1, and the angular velocity is 1, the equations of motion in the rotating frame are:
Let $\mu = m_2 / (m_1 + m_2)$ be the smaller body's mass fraction. Place $m_1$ at $x = -\mu$ and $m_2$ at $x = 1 - \mu$. Define the distances from the test particle to each primary:
$$ r_1 = \sqrt{(x+\mu)^2 + y^2}, \quad r_2 = \sqrt{(x-1+\mu)^2 + y^2} $$The full equations of motion in the rotating frame, including centrifugal and Coriolis terms, are:
$$ \ddot x - 2 \dot y = x - \frac{(1-\mu)(x+\mu)}{r_1^3} - \frac{\mu(x-1+\mu)}{r_2^3} $$ $$ \ddot y + 2 \dot x = y - \frac{(1-\mu)\, y}{r_1^3} - \frac{\mu\, y}{r_2^3} $$The terms that look like $-\!(\!\cdot\!)/r^3$ are gravity from each primary. The bare $x$ and $y$ on the right are centrifugal. The $-2\dot y$ and $+2\dot x$ on the left are Coriolis — they only matter when the body is moving in the rotating frame. Together these reduce to a fairly clean form.
An equilibrium in the rotating frame is a place where a body sitting still ($\dot x = \dot y = 0$, $\ddot x = \ddot y = 0$) stays still. With the time-derivative terms zeroed out, the right-hand sides of both equations must vanish. This is just a 2D root-finding problem and there are exactly five solutions, called the Lagrange points after Joseph-Louis Lagrange, who worked them out in 1772.
For Sun-Earth, with $\mu = 3 \times 10^{-6}$ (Earth is a millionth the Sun's mass), L₁ and L₂ both sit about 1.5 million kilometers from Earth — L₁ on the sun-ward side, L₂ on the anti-sun side. That's only about four times farther than the Moon, despite being a delicate three-body equilibrium. NASA parks real spacecraft at both: the SOHO solar observatory at L₁, the James Webb Space Telescope at L₂.
For Earth-Moon, $\mu = 0.0123$ and the Lagrange points are spread out: L₁ is 326,000 km from Earth (about 84% of the way to the Moon), L₂ is 449,000 km from Earth (about 65,000 km past the Moon), L₃ sits opposite the Moon at almost exactly Earth's distance from the barycenter. Various proposed lunar gateway and relay station concepts target L₂ specifically because it has continuous line-of-sight to the lunar far side.
L₄ and L₅ are real too, and stable enough that they collect debris. Jupiter's L₄ and L₅ in its orbit around the Sun host the famous Jupiter Trojans — over ten thousand asteroids, more than the rest of the asteroid belt combined, leading and trailing Jupiter by 60° on its orbit.
An equilibrium is just a balance point. What matters in practice is what happens if you perturb a body slightly off the point — does it drift back, or fly away? The collinear and triangular Lagrange points behave very differently:
The stability of L₄ and L₅ has a curious origin. The Coriolis force is what saves them. Without Coriolis, L₄ and L₅ would be hill tops in the effective potential — clearly unstable. But the moment a perturbed body starts to slide off, it picks up velocity, Coriolis kicks in perpendicular to that velocity, and the body curves back around the equilibrium point. It's a dynamical stability rather than a potential-well stability, and it's the only example of its kind that shows up in elementary mechanics.
The view below is the rotating frame, looking down on the two primaries with $m_1$ on the left and $m_2$ on the right. Background shading shows the effective potential (darker = lower). The five Lagrange points are marked. Click anywhere to drop a test particle at zero velocity in the rotating frame — its trajectory is integrated by RK4 with the full CRTBP equations including Coriolis. Try clicking near each Lagrange point and watch what happens.
Drop a particle at L₁, L₂, or L₃. Within a few simulated orbits the particle drifts away. The further it drifts, the steeper the slope, the faster it accelerates — exactly the runaway you'd expect at a saddle point.
Drop a particle exactly at L₄ or L₅ (Earth-Moon system). It just sits there. The effective potential is at a local maximum but Coriolis stabilizes it.
Drop a particle near L₄ but not exactly on it ("Near L₄" button). Watch it loop in a tadpole orbit around the equilibrium point — slow, stable, often years per loop in real units. This is the Jupiter Trojan dynamics in miniature.
Switch to "Exaggerated" (μ = 0.05). This is past the stability cutoff. Now particles dropped near L₄ wander away. The transition between stable and unstable triangular Lagrange points happens right around μ = 0.0385 — surprisingly close to the Earth-Moon ratio, which is why some textbooks misstate Earth-Moon L₄/L₅ as unstable. (In the idealized CRTBP they're stable; in reality solar perturbations destabilize them on long timescales, so there's no permanent population.)
Across eleven modules we've gone from a cannonball arcing over a flat horizon (module 1) through the propulsion engineering that makes any of it possible (module 9 and module 10) to spacecraft loitering at the equilibrium points of a chaotic three-body system. The thread tying it all together is just two ideas: Newton's law of gravity (an inverse-square force between every pair of masses) and conservation of energy (which gave us vis-viva, the workhorse of module 3 onward). Everything else — circles, ellipses, hyperbolae, Hohmann transfers, rocket equation, Trojans, JWST's parking spot — fell out of those two ideas.
If you want to see the curriculum's own destination, head over to SatView: a real-time 3D viewer that takes the orbital elements you met in module 6 and propagates them with Kepler's equation from module 4 to draw thousands of actual satellites doing exactly what this curriculum has been describing.