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.

Usage
Is the solar system stable? (Archive) Scott Tremaine University of Toronto and Institute of Astronomy, Cambridge

In the above paper Scott Tremaine performs an analysis of the Solar System. It is seen that this analysis uses Symplectic Integration techniques:



Related Examples


 * Symplectic Integrators For Solar System Dynamics by Scott Tremaine
 * High precision symplectic integrators for the Solar System by Jacques Laskar & Co.
 * Symplectic Maps for the N-body Problem with Applications to Solar System Dynamics by Matthew Jon Holman

New Scientist Article
A New Scientist article titled The World of Symplectic Space (Archive) provides a background and history of this subject, admitting that Newton's Laws of Motion cannot simulate the Solar System or the n-Body problems. This workaround using Symplectic methods which preserve geometry is necessary.

''Conventional calculations of the future of the Solar System quickly degenerate into disarray as computer errors build up. Symplectic integration could save the day''

Addendum
As explicitly admitted, Newton's Laws cannot simulate the Solar System. Exotic methods are required to preserve the area and geometry of an orbiting system.

One researcher states "Unfortunately, this beautiful theory does not apply to realistic solar system models".

New Scientist tells us that the methods cannot be used for prediction positions of bodies: "The result does not predict the exact motion of our Solar System, but it does provide useful information about it. For example, astronomers can work out Pluto’s distance from the Sun in a billion years’ time, but not which side of the Sun it will then be on."

All of this tells us that these methods are not realistic models of orbiting systems which can be used for prediction, and are really niche academic frivolities. Once again we are faced with the resounding message that the accepted laws of gravity and motion are not, and cannot, be used to simulate the Sun-Earth-Moon system or the Solar System.