Suppose that we have system of N astronomical objects, who is moved by gravity forces. Generally this problem is called N-body problem and its well known that starting from N=3, N-body problem can be solved only for special cases.
This work is covering more narrow situation called N-body choreography in which we are looking only for periodic solutions. So our question is: Which periodic orbits are possible for given N? I once discovered Moore's article and was amazed by how elegantly this problem can be solved with variational principle so I decided to implement it and visualize process of numerical learning. Further I will first briefly cover underlying theory and show animated results I achieved.
This work is an implemetation of method, proposed in article "Braids in classical gravity" by Cristopher Moore, 1993. This project was prepared during my studying at Skoltech University at Moscow as final project for course "Scientific computing".
I will start from afar and describe what matematicians call a variational principle. There is a well-known task of finding extremas of a function. Generally speaking this task is to find such that
is an extremum. Here is quite similiar task: instead of
we will have functions and instead of regular function
we will have "second-order" function
that takes some function (just consider it as curve on coordinate plane) as an input and calculates one number as an output. Here comes variational task: find such curve
where
will be minimal. I will not cover how this problem can be solved, just check out this wikipedia page.
It was discovered that mechanical laws (e.g. Newtown laws) can be derived from variational principle. It turns out, that trajectory of a moving body is such a minimal curve in case we define as a special concept called action. Action
is an integral over whole body path of another concept called Lagrangian
, which is actually just kinetic energy minus potential energy. Let's look on formulas:
That's it. All you need to do is find such curve where action
is minimal and it will be a real path. That is called the Principle of least action.
The principle of least action is used to solve our problem. We start from some approximate trajectories and then comupute to make gradient descent step and finally find appropriate trajectory.
For complete formulas see following literature:
- "Braids in classical gravity" by Cristopher Moore, 1993
- "New Orbits for the n-Body Problem" by Robert J. Vanderbei
Let's see what we have
Further for each case there will be three images:
-
starting orbits
-
process of learning, on plot you can see two types of lines:
- solid lines are for optimized trajectories, so for most part of the time they don't represent real gravitational moving
- dot lines are real trajectories from given starting conditions (calculated with leapfrog method)
that mean that we can consider our solution valid when this two types of lines overlap with each other
-
resulting orbits and bodies moving along them
For example see the simpliest case of 2 bodies
