Modeling Orbits of Planets and Galactic Matter
Modeling orbital systems is part of the science of
astrophysics, it's important to find ways to accurately predict the movement of the
planets,
stars,
asteroids and other stellar matter in studying their evolution and making models to explain past and future behavior. But this is not a straightforward endeavor for a number of reasons. Foremost of which is the
n-body problem. It is impossible to solve systems of
differential equations where we are looking at the positions and other physical data of objects numbering more than two. At least, it is impossible to do so analytically. It is possible to create computer simulations using numeric approximation.
Methods such as Euler's and Runge-Kutta allow us to estimate the curve that is created by a set of differential equations. These Numeric approximation methods are acceptable when they are used to find the shape of curves and solutions to differential equations. But at certain points they break down, because they make a round-off error somewhere along the way. For mathematicians with lots of time on hand this is fine, the just decrease the step-size they were using, get a better picture, then approximate the answer.
However in the sciences those little roundoff errors may be inescapable, not because step-sizes are too large but because they aren't able to synchronize the calculations of all the numbers involved in the experiment or phenomena. For an example, when modeling the solar system with just the Sun and planets it may be possible to set the planets in motion for any length of time. The addition of the moon, on the other hand, quickly causes the program to fail. Why is this? Because the moon's direction is changing too quickly for the approximation. As a result the approximation quickly sends the Luna spiraling into the Earth. The reason this occurs is the fact that the Energy of the system is not kept constant or at least kept reined in, the result is that when the approximation finds an answer far enough off of the mark then suddenly the moon starts losing energy or gaining it. The Runge-Kutta Method doesn't care about energy, just position and velocity.
Symplectic Methods
Symplectic methods are approximation methods that take total energy into account when calculating position and velocity, because these values determine total energy according to the Hamiltonian. Such methods work under the assumption that energy is not being added to the system or taken away. Thus a Matlab program that uses this method would apply Runge-Kutta to the given data, but constrain the algorithm using the Hamiltonian.
Using the Hamiltonian improves numeric methods involving energy, which occur again and again, but in the case of stellar modeling this may not be enough to account for all movement, given the fact that gravitational information travels at the speed of light. So a star orbiting the Galactic Core isn't attracted to the Core's current position but the position that it was in the number of years ago that corresponds to the number of light-years that separated them at that time. Understandably these sort of methods require further changes.
Modeling Galaxies
There is a further problem to solve. The problem is that once you've created a method which takes millions or billions of particles and set them in motion, calculating the effects an entire galaxy has on each star in that galaxy, even if you are running your simulation in a huge server farm or a dedicated supercomputer, it will take days if not weeks for results to be made output. This is not an insurmountable problem, it's just that when we look at billions of things and have to make billions of calculations for each, or even half of them, we're going to take too long, the solution to this problem is to simply not do it. Instead we find it is easier to calculate the necessary data using an entirely different set of differential equations, ones which look at a mass of billions of particles and doesn't think about each one individually. Scientists look to fluid dynamics, that is they look at all the stars in the galaxy and see a sea of stars. With this in use it is possible to relatively quickly attempt the creation of models of galactic kinematics.
Theory and Hypothesis
It is actually theorized that if we model the galaxy using methods that conserve energy and general and special
relativity it is possible that the anomalous movement that led people to postulate the existence of
Dark Matter could be explained without wimpy particle halos.
Some good works on the subject:
Binney, Gerhard, et al. "Understanding the kinematics of Galactic Centre gas"
Cooperstock and Tieu. "General Relativity Resolves Galactic Rotation without Exotic Dark Matter"