There are times in physics when what seems like a simple extension to a problem causes the floor to drop out. The motion of three bodies due to gravity is one of those cases. For the two-body solution there is a modest differential equation and a solution that describes the motion as an ellipse, parabola or hyperbola. Adding a third body seems like “no big deal”. You might think that the math gets a bit messier so maybe the solution will not be quite as clean as a simple ellipse – but there should still be some kind of sensible solution. This is one of those cases where our optimism/intuition fails.

Newton’s work was published in the 1680s. It took other great mathematicians to get further. Euler found an unstable 3-body solution in the early 1700s. Lagrange found another unstable solution as well as some stable solutions in special cases for the 3-body problem. His work was published in 1772 – almost a hundred years later. [Lagrange also fundamentally changed physics with his *Mechanique Analytique*. Today huge parts of modern physics is still rooted in “the Lagrangian”].

Lagrange found a few special cases where three bodies could stay in a stable solution by making simplifications that result in what is called the “restricted three body problem”. In this case one body is “heavy”, the other has moderate mass and the third has no mass. As a concrete example think of the Sun, Jupiter and an asteroid in the same orbit around the sun as Jupiter. As a further simplification, assume Jupiter moves in a perfectly circular orbit. It is then convenient to think in terms of the position of the third massless body in relation to Jupiter i.e. in a co-ordinate frame that rotates around the sun at exactly the same rate as Jupiter. Lagrange found there were five points at which the third body experienced zero net force and could stay fixed in relation to Jupiter in this rotating frame.

This in itself was an interesting result. However these five Lagrange Points would only be really interesting if the massless body actually stayed in this relation to Jupiter even if it was bumped away from this zero force location i.e. if the motion around a Lagrange point is stable. It turns out that two of the five Lagrange points are stable but there’s a catch. The catch is that the Lagrange points L4 and L5 are stable only if Jupiter is not “too heavy” compared to the sun. Exactly how heavy? L4/L5 are stable as long as Jupiter is not more than 3.85 percent the mass of the Sun.

These points are of more than academic interest. This solution motivated a search for asteroids near these locations in Jupiter’s orbit and as the technology of telescopes improved the first such asteroid, Achilles, was found in 1906. There are now thought to be as many of these “Trojan” asteroids as there are asteroids in the belt between Mars and Jupiter. Other planets have such asteroids as well. Recently, more asteroids were found at the L4 point of Mars (http://arxiv.org/pdf/1303.0124.pdf).

What if the test particle is not quite at the stable Lagrange points L4/L5 or does not have quite the right velocity? Then the resulting motion is very interesting. This leads to orbital paths relative to Jupiter that get names such as tadpole or horseshoe orbits, due to their shape – you can get a sense of why these names arise by playing the levels in the Lagrange group in N-body. The image below shows oscillations about the L5 point in the N-body iPad app.

Oscillations about Lagrange points also have real-world implications. For example, the detection of a dust ring around Uranus puzzled astronomers since location of the ring could not be stable. It was proposed that there were “sheparding satellites” ^{1}. Subsequent discovery of two small moons, Ophelia and Cordillia by Voyager lend credence to this idea. ^{2}

Amazingly (and a testament to just how opaque the physics becomes when N=3) another *stable* three body solution for *equal* masses was found in the 1990s more than two hundred years after Lagrange. It was a result of considering topologies of paths. We’ll look at this fascinating solution and some of it’s unstable cousins in the choreography group of N-body levels.

One aside: In general relativity the step of going from one to two makes the floor fall out. In the case of Einstein’s general theory of relativity (GR) finding a solution for ONE body is really hard. The spherically symmetric solution was found by Schwarzschild in 1916 about a year after GR was published. It took until the 1960s before the axially symmetric one-body solution was found. There are no two body solutions corresponding to a simple orbit that can be written in terms of modest mathematical functions. This is part because in GR the idea is that gravitational interactions do not operate via a force but by defining the shape of the space they are part of. As bodies move around each other there are ripples in space-time (waves) that need to be taken into account. This makes things very messy. Numerical modelling and simplifications can be employed and things can be learned – but there is not a tidy mathematical description.

Notes:

- Goldreich and Tremaine (1979) Nature
**277**97-99 ↩ - See the discussion in 10.5.3 of Solar System Dynamics ↩