Symplectic Integrators

As result of the famous Three Body Problem mathematicians have opted to create special algorithms for use in multi-body problems as a method of keeping the system stable. Symplectic Integrators are special geometric integrators which preserve the geometry of an orbiting system.

Wikipedia defines Symplectic Integrators as:

Integrator Comparison
Computing the long term evolution of the solar system with geometric numerical integrators (Archive)


 * Abstract



The standard algorithms produce 'wrong solution' because the Moon is ejected from its orbit. A different algorithm is necessary to keep it together, and produces the 'correct' behavior. The paper describes that the algorithm which keeps it together is the symplectic integrator.



A diagram shows a comparison:


 * Sun-Earth-Moon-Integrators.png

A history of the n-body problem is given:


 * History - Is the Solar System Stable?




 * Conclusion



A special integrator, called symplectic integrators, which preserves the energy and prevents the body from escaping is used, and is therefore 'more suitable'. Does this sound like a full clean simulation of gravity?

Always Returns Stable Conditions
The following is a paper which looks at symplectic and non-symplectic simulations and says that symplectic integrators always give stable conditions regardless of the perturbations which affects the body:

Numerical Integration Techniques in Orbital Mechanics Applications (Archive)

We read that the non-sympletic integrators do not give stable solutions, while symplectic integrators are "always able to return to stable conditions after perturbations".

Does an algorithm which always returns to stable conditions sound like one which accurately represents reality?

Description
The purpose of the symplectic integrator is to preserve the area or geometry of the phase space.

University of Rochester PHY411 Lecture notes Part 7 – Integrators (Archive)

Different trajectories:



Further down in the document:



...



Phase Space vs. Position Space
It has been argued that these definitions apply only to "Phase Space," and that Phase Space is entirely different than the normal Position Space that we know which uses the x, y, and z coordinates. It has been argued that Phase Space does not contain the Position Space coordinates.

In truth, Phase Space is merely Position Space with additional dimensions. We read a definition from Physics for Degree Students B.Sc Second Year (Archive):


 * 10.4 Phase Space



Scrolling up to section 10.2 we read the definitions of Position Space, verifying that it is the normal x, y, z position coordinates:


 * 10.2 Position Space

Thus, if the geometry of Phase Space is preserved with a Symplectic Integrator, the geometry of Position Space is also preserved.

Figure Eight Example
Compare this figure eight three body problem in Phase Space and Position Space:

https://demonstrations.wolfram.com/PlanarThreeBodyProblemInPhaseSpace/ (Archive)



Compare the above to the Figure Eight Three Body Problem solution in Position Space, plotted on the x and y.

https://www.researchgate.net/publication/253681041_Dynamics_and_stability_of_multiple_stars (Archive)



Plotted on the x and y, and looks surprisingly like the Phase Space version.

It looks the same because it is the same. Phase Space is merely Position Space with detail in additional dimensions which represents momentum. The shape of an orbit represents geometry, as well as energy loss/gain.