Symplectic Integrators

As result of issues with the Three Body Problem mathematicians have opted to create special algorithms for use in multi-body problems as a method of keeping the system stable. Usually associated with the Hamiltonian and KAM theories, Symplectic Integrators are special geometric integrators which preserve the geometry of an orbiting system.

Wikipedia defines Symplectic Integrators as:

Sun-Earth-Moon System
Computing the long term evolution of the solar system with geometric numerical integrators (Archive) By mathematicians Shaula Fiorelli Vilmart (bio) and Gilles Vilmart (bio)


 * 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.

Figure 7 from the paper shows a comparison between a non-symplectic and symplectic integrator:


 * The Sun-Earth-Moon System


 * Sun-Earth-Moon-Integrators.png



The paper describes that the algorithm which keeps it together is the symplectic integrator.




 * Initial Data




 * References


 * [8]Observatoire virtuel de l’IMCCE, Portail système solaire, Miriade ephemeris generator, Observatoire de Paris & CNRS (2016), http://vo.imcce.fr/webservices/miriade.

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?


 * Source Code

The above paper includes the source code for the above Sun-Earth-Moon System, for use with the free open source software Scilab, where the integrators can be tested.


 * Sun-Earth-Moon.gif

Sun-Jupiter-Saturn System
Another paper states the same for the Sun-Jupiter-Saturn system:

Initial Data



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 a system to stable conditions after perturbations sound like a legitimate reflection of bodies operating under Newton's laws?

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)

Figure 4 shows different trajectories:


 * Orbit Trajectories.png

Further down in the document:


 * Symplectic Description.png




 * Symplectic Description 2.png

We read above that symplectic integrators are designed to preserve the geometry of phase space.

Solving the Hamiltonian
A paper Symplectic integration of hierarchical stellar systems by H. Beust describes that the symplectic integrator does not actually solve the real Hamiltonian of the problem:

https://www.aanda.org/articles/aa/full/2003/12/aa3133/aa3133.right.html (Archive)

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)


 * Three Body Phase Space.png

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)


 * Three Body Real Space.png

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, concluding that the Solar System will be stable for millions of years. It is seen that this analysis uses Symplectic Integration techniques:


 * Tremaine-Symplectic.png

Related Examples


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

New Scientist - The World of Symplectic Space
A New Scientist article titled The World of Symplectic Space (Archive) by Robert McLachlan, Ph.D. (bio), 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. It is necessary to use a workaround using Symplectic integration, which preserves area and geometry.



Solar System Integrator Comparison
Below we find a simulation of the outer planets of the Solar System, showcasing the Forward Euler method versus the Symplectic Euler method.


 * Numerical_Integration_of_the_Solar_System_2.mp4
 * (Archive 1 2)


 * Description:

We see in the above video that with the Euler Forward method that Jupiter is ejected from the Solar System after a single orbit around the Sun. Saturn is thrown beyond Neptune, &c., as the Solar System quickly degenerates.

Forward Euler Descriptions
Wikipedia states that the 'Forward Euler method' is also known as the 'Euler method':



Descriptions from Science Direct sources state:



Source Files
The author provides steps for reproducing his plots with MATLAB in the description of his linked associated video, which shows a successful Störmer–Verlet simulation of the Solar System:


 * ''This is a demonstration of the Stormer-Verlet method applied to the outer solar system consisting of Jupiter through Pluto. The computation was carried out in Matlab with a time step of 200 days, a final time of 200,000 days, distances in astronomical units (AU), and masses normalized by the Sun's mass.  The masses of Mercury through Mars were lumped together with the Sun.  The initial conditions for position and velocity along with the scaled masses were obtained from pg. 13-14 of the 2nd Ed. of "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations" by Hairer, Lubich, and Wanner.


 * To reproduce this plot, download the main script and functions linked below and run the main script with the functions either in the same folder as the main script or somewhere else in Matlab's search path (for Windows users C:\Users\YourUserName\Documents\MATLAB). The only variable that needs editing in the script is "method = methods{index}", where "index" should be the number corresponding to the numerical method you want to use from the list given below.


 * The numerical methods provided include:
 * 1) Forward Euler
 * 2) Runge-Kutta 23
 * 3) Runge-Kutta 45
 * 4) Symplectic Forward Euler
 * 5) Stormer-Verlet


 * Main script:
 * Solar_System_Example.m


 * Functions:
 * solarSysDiffEQ.m
 * solarSysHam.m

Störmer–Verlet is Symplectic
It is seen that the Störmer–Verlet is a symplectic method.

Paper: Qualitative study of the symplectic Störmer–Verlet integrator

Abstract:

Quotes
Symplectic integrators are used in other areas in science. From the abstract of a particle physics paper (Archive) from the 6th International Particle Accelerator Conference we read that they are required to keep simulations stable:

Symplectic integratos provide approximations of mechanical systems. From the abstract of a 2016 paper from Physical Review:

From the abstract of Hamiltonian Approximants for Symplectic Integrators we see:

Numerical Solutions
From the abstract of Symplectic Runge-Kutta and related methods: recent results (Archive):

See Numerical Solutions - The available solutions for celestial systems do not fully simulate gravity. For instance, the following is a symplectic integrator which breaks gravity down into multiple two body problems, rather than using a single multiple body problem:

https://academic.oup.com/mnras/article/452/2/1934/1069988 (Archive)



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

On KAM theory New Scientist states "While the computation will still, inevitably, accumulate errors, the KAM theorem guarantees that they will not nudge a planet into an impossible orbit.", while another researcher remarks regarding KAM theory - "Unfortunately, this beautiful theory does not apply to realistic solar system models". One paper concludes that "the symplectic integrator, by keeping the Hamiltonian constant, is able to bound the error and prevent it from growing substantially, always able to return to stable conditions after perturbations."

New Scientist tells us that that the simulations are not built on the laws of motion that apply in our familiar three-dimensional space, that the conventional models "quickly degenerate into disarray", and that these alternative geometric methods discussed cannot be used for prediction of the 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." Which is interesting since any direct and real simulation of motion would inherently know that Pluto makes an orbit every 248 years around the Sun, regardless of the author's downplay of 'a billion years'. This is also a contradiction of the popular claims that the Solar System can be truly simulated with Newton's laws out to millions or billions of years, or even for a shorter period of time barring the special geometry preserving methods which keeps it together.

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 — the end result of a submission to the Three Body Problem and science's desperate attempt to get invalid laws and an unworkable system to work.

Instead of a working model based on gravity and the laws of motion we are presented with a system based on "geometric laws". The truth is that if Newton's laws could keep the Solar System together he would have demonstrated that himself in the 1700's, rather than concluding that divine intervention keeps the Solar System stable, and which this geometric approach is the ultimate manifestation of. Newton's own conclusion that the Solar System is unworkable under his laws was true then and remains true today. We are once again faced with the reality 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.