Numerical verification of the microscopic time reversibility of Newton’s equations of motion: Fighting exponential divergence
- by Simon F. Portegies Zwart Tjarda C. N. Boekholt 2018 - Article review

This document contains article review "Numerical verification of the microscopic time reversibility of Newton’s equations of motion: Fighting exponential divergence" - by Simon F. Portegies Zwart Tjarda C. N. Boekholt written 2018
To order to read the article select:https://arxiv.org/pdf/1802.00970.pdf

• The text in italics is copied from the article.
  
• Immediate followed by some comments
  

Contents

• 1. Introduction
• 2. Results for the Pythagorean 3-body problem
• 3. Discussion

Reflection

• Reflection 1 - Numerical Simulation
• Reflection 2 - Time reversibility

Abstract

Numerical solutions to Newtons equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible.

See: Reflection 1 - Numerical Simulation . The subject is any physical process consisting of n objects. For example: a comet which revolves around the Sun. Such a physical process is irreversible. In the case of a comet during each revolution the comet loses matter. The physical process is not considered chaotic. For example: A roulette is not studied.

This is due to the exponential growth of errors introduced by the integration scheme and the numerical round-off in the least significant figure.

This type of problem exists if the mathematical system is 'intrinsic' unstable. For example: when your simulated system consists of 8 objects of equal mass, around a circle, at 45 degrees from each other.

This secular growth of error is sometimes attributed to the increase in entropy of the system even though Newton’s equations of motion are strictly time reversible.

The error growth is a mathematical issue and not a physical issue like entropy.
Newton's equations equations can considered as 'time' reversible. The proces studied is not.

However, time reversible algorithms may be used to find initial conditions for which perturbed trajectories converge rather than diverge.

This is a strange strategy because initial conditions are considered physical 'fixed' i.e are calculated as a result of physical observations.

The ability to calculate such a converging pair of solutions is a striking illustration which shows that it is possible to compute a definitive solution to a highly unstable problem.

When the system studied is (physical) unstable the simulated solution should also be unstable.

1. Introduction

General analytic solutions to problems in Newtonian dynamics can only be achieved for a single particle, N = 1, or for N = 2.

The most stable physical system is a binary star system, consisting of exactly 2 stars. No planets are involved. From a physical point of view those systems are not realistic i.e. interesting.
A whole different situation arises when both objects are Black Holes. Such a system is supposed to be not stable, because both BH's are supposed to merge.

Families of periodic solutions exist for N > 2, and in particular the parameter-space search of has succesully identified more than 1000 new periodic solutions to the restricted 3-body problem, suggesting that the number of such solutions is interminable.

Both the books: "Astronomy with your Personal computer" by Peter Duffett-Smith and "Astronomical Algorithms" by Jean Meeus use analytical solutions to simulate the trajectories (positions) of the planets. Both book are excellent but they don't have the accuracy to demonstrate the forward movement of the perihelion of the planet Mercury. They require a different strategy.

For all other solutions approximate methods have to be employed.

I understand that for n=4 no analytic solution exist. The use of difference equations seems to me the only strategy

Families of periodic solutions exist for N > 2, and in particular the parameter-space search of [6] has succesully identified more than 1000 new periodic solutions to the restricted 3-body problem, suggesting that the number of such solutions is interminable.

Page 3

2. Results for the Pythagorean 3-body problem

3. Discussion

A simulation is a numerical description of the physical reality. The central part is physical law i.e. a mathematical description of the system studied. For example: the planets of the solar system. This type of simulations are called N Body simulations. In the case of N=3 we can study the following 3 objects:

• The Sun and the two planets: Mercury and the Earth.
  This simulation is interesting because it shows a relativistic component
• The Earth, Jupiter and a satelite.
  This simulation is used to simulate the sling shot effect. The slingshot effect is very sensitive about the initial conditions.

  When you want to perform any simulation three things are very important:

  • A computer program which excutes an algorithm or mathematical law. In this case that is Newtons law. In that case because we want to predict the positions of three objects this are 3 times 2 equations. One to calculate the speed and one to calculate the position. There are many different ways to calculate these parameters. The most important issue is the number of previous positions involved.
  • Observations. Observations are the positions of the objects considered in the past. Using these observations the parameters of the equations are calculated. In this case the masses of the objects. All observations are involved.
  • Initial conditions. In summary, the position and velocity at the start of the simulation of each object. These two values have to be calculated as a function of all observations.

  This strategy is also used in the following two programs:

  • "VB2019 BHmerger". The program VB2019 BHmerger demonstrates the issues involved with BH mergers.
    For more detail see: VB2010 BHmerger operation
  • "VB2019 Sagittarius". The program VB2019 Sagittarius demonstrates the 10 large stars around the BH.
    For more detail see: VB2019 Sagittarius

Reflection 2 Time reversibility

Time reversibility is typical a mathematical problem, but not a physical problem. In physics neither time nor time reversibility exists. In physics you describe a process by using a differential equation and this differential equatial equation can be used as a model for a difference equation which can be used as a building block as part of the computer program to simulate the physical problem.
Such a simulation can be used to predict the the future by running the time in the forward direction starting from the initial conditions at the beginning of the simulation.
Such a simulation can be used to predict the the past by running the time in the backward direction starting from the initial conditions at the end of the simulation.
That is what I think is the (mathematical) strategy.
The problem is that Newton's Law is a too simple description of the physical reality. Newton's Law assumes that gravity acts instantaneous, in reality the speed of gravity (waves) has a final speed. That means each object is influenced by a 'local' gravitational field at present, but that field is created by the positions of the objects in the past.
When you assume that gravity acts instantaneous, then there is no difference between both. But in case of the movement of the planet that is not the case, meaning Newton's Law has to be modified to include the speed of gravity. A different strategy is to use the General Theory of Relativity.

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