The Multiple Body Problem
When the Theory of Gravity Collides with Chaos Theory
The n-body problem - which defines the gravitational attraction of multiple bodies - has plagued mathematicians and physicists for more than a century.
Isaac Newton published his rather simple equation for universal gravitation in 1687. As far as equations which successfully model the behaviors of the natural world go, this equation is strikingly simple:
F = G(m1 m2)/d²
It is a simple inverse square law, meaning that the force of gravity between two objects decreases in an inverse proportion to the square of the distance between them.
The Three Body Problem
Simple as this may seem, what happens when a third object is added into the mix? How, then, does the force of gravity effect the three of these objects as they relate to each other?
Could it be as simple as just adding another variable to Newton’s equation?
Of course not.
The three body problem, in fact, turns out to be so very difficult that it has stumped mathematicians and physicists alike for centuries now – ever since Newton’s original equation first arrived on the scene.
To date, there has been much progress made in this particular area. Individual cases of the three body problem have been solved, though this is not the end of the story. In order for mathematicians to be able to say that they’ve closed the book on a problem – completely solved it – they have to find a universal equation.
Finding only the solutions to specific problems are very important steps, indeed, but not the final answer that might be hoped for.
Still, these specific cases are good to look at, to give one a better perspective of what the three-body problem entails
First, there is the “Circular restricted” three-body problem, which makes the bold assumption that one of the three bodies is of a negligible mass compared to the other two, and thus of only very minimal effect on the orbit as a whole.
Second, there is the idea of “Lagrange Points.” These are very precise areas of equilibrium lying in between two massive bodies. A third object, sitting precisely at this point, would be equally effected by both of the other bodies, and thus cause no disturbance in the system as a whole.
The Lagrange points formed by the Earth and the Moon and the Earth and the Sun have been precisely calculated, and serve as invaluable places to potentially place such things as satellites and space telescopes, for it would only take a minimum amount of energy in order to keep them in place.
The N-Body Problem
As difficult as the three-body problem is, it is almost indescribably simple compared to the problems created by adding further bodies to the equation.
The n-body problem (which simply means that the number of bodies is a variable, and can be any number imaginable), has proved to border on the impossible.
The problem has proved to be so difficult, in fact, that in the late 19th century, King Oscar II of Sweden offered a substantial prize to the person who could finally solve the King’s own specific version of the problem (a problem finally solved for the 3-body problem by Karl Sundman in 1912, though not for a universal n-body problem).
Multiple Bodies and Chaos
What is the reason for these difficulties?
Simply put, anything more than two bodies in a system, and the solution can very quickly turn into one best exemplified by chaos theory. Multiple bodies cause perturbations on each other which, while perhaps small and miniscule at first, become compounded as more time passes, quickly multiplying the variety of possible outcomes by an exponential degree.
This being the case, even in a case such as the circular restricted 3-body problem, the tiny perturbations caused by the negligible mass of the third body would cause, in the extremely long run, enough change to make predictions far into the future impossible, in an astronomical version of the “butterfly effect.”
For a remarkable visual example of how tiny changes can effect the orbits of various massive bodies, visit this resource.
So, for example, as the moon orbits the sun, the moon’s effect on the sun is never accounted for in equations, because it’s size is so small in comparison. The farther one looks into the future, though, the more it becomes clear that any problem with three or more bodies will eventually devolve into chaos, where an entirely new type of mathematics must be used in order to find the beautiful order in all that noise.
References:
“Collection of Remarkable Three-Body Motions.”
