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Introduction
I’m reading “three bodies” (santai)vol1, vol2, vol3.
Three Bodies is a full-length science fiction novel by Chinese science fiction writer Liu Cixin, serialized in the Chinese science fiction magazine “Science Fiction World” from May to December 2006, and published in book form by Chongqing Press in January 2008, as part of the “Earthbound” trilogy.
This novel is a blockbuster work, containing five book-length stories with a total of 1,963 pages. The Hyperion series by Dan Simmons< is the most famous full-length science fiction novel, which is published in eight books and has a total of nearly 3,000 pages,
While the spiritual world is depicted based on ideas such as human computers and virtual worlds, on which the movie Matirx is based, “Three Bodies” is an accurate, logical and rigorous depiction of known astronomy, physics, chemistry, mathematics and engineering techniques, and theoretically possible ideas backed by this scientific knowledge. Hard science fiction” is at the center of the story. Writers of hard science fiction include Arthur C. Clarke , Isaac Asimov,(mentioned in “Foundation“), Greg Egan,and James P. Hogan.
In 2015, Three Bodies was the first book by an Asian author to win the Hugo Award for Best Full-length Novel. That same year, 2015, Facebook CEO Mark Zuckerberg chose “Three Bodies” as one of the books in his “A Year of Books” project, in which he recommends a book every two weeks. Zuckerberg recommends the book as “a pleasant break from the heavy economics and social science books I’ve read recently.”
Also on January 16, 2017, then-U.S. President Barack Obama revealed himself to be a fan of the “Three Bodies” series in an interview with The New York Times, where he said, “It was very imaginative and really interesting. Reading about the fate of the vast universe made me think that the problems we face in Congress every day are fairly trivial and nothing to worry about.
Furthermore, American film director James Cameron revealed himself to be a devotee of the Three Bodies trilogy during a meeting with Liu Cixin. Japanese game designer Hideo Kojima is also a fan of the “Three Bodies” series, and wrote in his recommendation for the Japanese edition, “It is a miraculous ‘super-ton-demo SF’ that was born from the exquisite Lagrangian balance of universality, entertainment, and literature in the gravity of the ‘Three Bodies.
The main theme among these three bodies is the three-body problem. In this article, I would like to discuss the three-body problem.
The Three Body Problem
The three-body problem is a classic problem in mechanics and an important issue in celestial mechanics. The three-body problem refers to the interaction and motion of three masses (or celestial bodies) under the action of gravity.
Specifically, in the three-body problem, three masses (or celestial bodies) interact gravitationally, and these masses can be celestial bodies such as star bodies, planets, satellites, or other objects with mass. The key to the three-body problem is to find the motion trajectory of the masses in the gravitational field, but due to the complexity of the gravitational interaction, there is no exact solution to the three-body problem, and even in the general case, the three-body problem requires complex computational methods and many computational resources for numerical analysis.
The simplest case of the three-body problem is when the masses are equal and the initial condition is symmetry, which is called the isomass three-body problem. Even in the isomass three-body problem, the motion trajectories of the masses can be stable or chaotic. If this is extended to a more general case, the motion trajectories of the masses are even more complex, with periodic, stable or chaotic trajectories appearing.
The study of the three-body problem is important for understanding the fundamental laws of celestial motion and celestial mechanics, and has been widely applied in the fields of celestial mechanics, space engineering, and spacecraft orbit design. The three-body problem has also attracted the interest of many scientists and mathematicians and has produced important mathematical and physical results.
Why can’t we solve the 3-body problem?
As mentioned above, the three-body problem becomes a problem of accurately predicting the interaction of three massive bodies (e.g., the sun, earth, and moon). The problem can be solved based on Newton’s equations of motion, but no analytical solution exists. Below we discuss the reasons why the 3-body problem cannot be solved.
- Nonlinearity: The three-body problem is a nonlinear problem. That is, the interaction of the bodies follows the law of gravity, and the changes in the positions and velocities of the bodies are intertwined with each other. This nonlinearity makes it very difficult to find an analytical solution.
- Unpredictability: Three-body problems can exhibit chaotic behavior. Even small changes in the initial conditions can cause significant differences in the results, and the impact of small changes on the long-term results is unpredictable, making it impossible to obtain an exact solution.
- LACK OF ANALYTICAL SOLUTION: In the analytical present of the three-body problem no solution exists. Newton’s equations of motion have analytical solutions for the interaction between two bodies, but no analytical expression exists for three or more bodies. Therefore, a numerical or approximate approach is essential.
Thus, the three-body problem is considered unsolvable, and the practical aspect of the problem is to use numerical simulations and approximate methods to predict the orbits of celestial bodies under specific initial conditions.
Approximate solution of three-body problems
The following approaches are used to approximate the solution of the three-body problem.
- Use of the center-of-gravity system: The center-of-gravity system is a coordinate system whose origin is the center of gravity of the position vectors of the three quality points. In this coordinate system, the sum of the position vectors of the three quality points is a zero vector. In the center-of-gravity system, only the motion of the center of gravity is considered, and the other two quality points are approximated as going around the center of gravity. This may allow a three-body problem to be treated as a two-body problem.
- Decomposition into a two-body problem: Among the three masses, the interaction of two masses is analyzed as a two-body problem, and the remaining one mass is considered passively from the average interaction of those masses. This approximation allows for a simplified analysis of the three-body problem, but is limited to cases where the interactions of the motions of the masses are equal.
- Mean Field Approximation: In the mean field approximation, each mass point is assumed to be affected by the average of the other masses. In other words, each mass point is considered to be in motion in a gravity field averaged over the collective effects of the other mass points. In this approximation, the three masses are treated as independent masses and the motion of individual masses can be analyzed.
- Numerical Methods: Numerical methods are an approach to obtaining numerical solutions to the three-body problem using methods for solving differential equations numerically. Typical methods include differencing Newton’s equations of motion to obtain numerical approximate solutions at each time step and simulation methods (e.g., Monte Carlo methods, N-body simulations).
Among these approaches, we will discuss the mean-field approximation and the Monte Carlo approach, which are also used as algorithms for machine learning with stochastic models.
Approximation of three-body problems by mean-field approximation
The mean-field approximation, also described in “Calculating the Surrounding Probability Distribution – Mean Field Approximation” is also the algorithm used in Bayesian estimation. In this approximation, each mass point is assumed to be affected by the average of the other mass points, and each mass point is considered to be in motion in a gravity field averaged over the collective effects of the other mass points.
The following is the general procedure for solving the three-body problem by the mean-field approximation.
- Set initial conditions: The initial positions and velocities of the three masses are given.
- Set the time: Determine the time range and time steps to be solved.
- Main loop: Repeat the following steps for each time step.
- Calculate the average field at the position of each mass point. This will be the average gravity field affected by the positions of the other two masses.
- Calculate based on the relationship between the position and mass of the mass point.
- Solve the equations of motion for each mass point. The mass point is in motion according to the mean-field gravity from the other two masses. The equations of motion are obtained using Newton’s laws.
Update the position and velocity of the masses. This will update the position and velocity of the masses to new values based on the time step.
- Obtain analysis results: After completing the main loop, obtain the position and velocity results for each masses point.
In the solution using the mean-field approximation, each mass point is subject to the collective effect of the other mass points as it moves, so the calculation can be simplified by averaging the interaction between the mass points. However, since the mean-field approximation is not an exact solution, caution should be exercised, especially when long-term predictions or high accuracy is required.
Analysis of three-body problems using Monte Carlo methods
The Monte Carlo method, described in “Markov Chain Monte Carlo (MCMC) Method and Bayesian Estimation” is a method of numerical analysis using stochastic techniques and is used to analyze stochastic models such as Bayesian estimation and can also be applied to the analysis of three-body problems. The novel “Three Bodies” describes solving the three-body problem using this Monte Carlo approach. The following is a description of the Monte Carlo approach to the analysis of the isosceles three-body problem.
- Generation of random initial conditions: The Monte Carlo method generates random initial conditions. This is done by setting random values for the positions and velocities of the masses. The generated initial conditions must satisfy the symmetry and constraint conditions of the system.
- Run a numerical simulation: Using the generated initial conditions, run a numerical simulation. Numerical integration methods (e.g. Runge-Kutta method) are used to calculate the time variation of the position and velocity of the masses. Starting from the initial conditions, the motion of the masses is calculated at regular time intervals.
- Perform the analysis: Analyze the results of the numerical simulation. Since the Monte Carlo method generates a large number of initial conditions and performs numerical simulations for each of them, the results obtained are statistical in nature.
By performing these analyses, the stability, characteristic behavior, and existence of chaos of the trajectory can be investigated. For example, the shape and characteristics of the motion of the trajectory can be compared for each initial condition, and statistical analysis can be performed.
The Monte Carlo method is useful as a method for statistically analyzing the complex behavior and interactions of three-body problems, but even when the Monte Carlo method is used, its statistical nature requires a sufficient number of samples and analytical rigor, and care must be taken to ensure the reliability and accuracy of analytical results.
Finally, we discuss the isomaterial three-body problem and its approach, which is an analyzable special solution of the three-body problem.
Solving isomeric three-body problems
The isomass three-body problem has special properties when the masses between the masses are equal, and it has analytical solutions under certain conditions. Some of them are described below.
- Lagrange’s Solution: Lagrange proposed a solution to the equimassive three-body problem in 1772. In this solution, the three-body system is represented by an appropriate coordinate system, and the relative positions and velocities of the points are parameterized. It is known that at a special configuration of points called Lagrangian points, which is obtained from the above, the mass points are in a state of equilibrium, and these Lagrangian points often appear in science fiction such as Mobile Suit Gundam.
- Hill’s Equations: Hill’s equations are a way to express the isomass three-body problem in the form of differential equations. Hill’s equations are difficult to solve analytically, but numerical analysis and approximate solutions can be used to predict the motion of the masses.
- Poincaré Section: The Poincaré section is a method for observing the chaotic behavior of the isomaterial three-body problem. This method allows us to understand the dynamics of the system by observing the motion of the masses on a particular plane and analyzing orbital intersections and characteristic patterns.
No analytical solution method exists for the general isomaterial three-body problem. In such cases, numerical analysis or computer simulation is commonly used to numerically solve for the motion of the masses, and numerical methods used include Newton’s method as described in “Overview, Algorithm and Implementation of Newton’s Method“., Runge-Kutta method, and direct method (Direct method).
The above three solution methods are further described below.
Equivariant three-body problem with Lagrangian solution method
The Lagrangian solution method is an important algorithm for solving optimization problems in machine learning, as described in “Dual problem and Lagrange multiplier method” etc. As one of the methods for solving isomaterial three-body problems, it is a tool for analyzing equilibrium states at points with special arrangements by parameterizing the relative position and velocity of the masses. It is a tool for analyzing the equilibrium state at a special configuration of points by parameterizing the relative positions and velocities of the masses. The following is a description of the basic ideas behind the Lagrangian solution of the isomaterial three-body problem.
In the equimassive three-body problem, it is assumed that the masses of the three mass points (A, B, and C) are equal. This increases the symmetry of the system and the existence of equilibrium points and special orbits in certain configurations.
First, Lagrangian coordinates are introduced to parameterize the relative positions of the masses A, B, and C. In Lagrangian coordinates, the center of gravity of mass points A and B is the origin, the vector from A to B is x, and the vector from C to the center of gravity is y.
Then, to represent relative velocities, the time derivative of x is Vx and the time derivative of y is Vy.
With these parameterizations, the isomaterial three-body problem can be expressed in the Lagrangian coordinate system. The Lagrangian solution method then analyzes the stability and special trajectories of the system by looking for equilibrium points in a particular configuration.
Of particular importance here will be points in a special configuration called Lagrangian points. The Lagrangian point is the point at which the masses are in equilibrium under the effect of gravitational attraction, and is an important indicator of the stability of the system. For example, in the earth-moon system, there are Lagrangian points (L1, L2, and L3) centered at the centers of gravity of the earth and moon.
The analysis of the isomass three-body problem by the Lagrangian solution method is more complicated and difficult than general analytical methods, and in particular, further approximations and numerical analysis are necessary because the isomass constraint is often not valid for actual celestial systems. However, the Lagrangian solution method for the isomass three-body problem is useful in applications such as celestial mechanics and spacecraft trajectory design, especially in orbit stabilization and resource optimization using Lagrangian points.
On the Isogeneous Three-Body Problem by Hill’s Equation
Hill’s equations, which are used to describe the relative motion of masses, provide a way to express the isomass three-body problem in the form of differential equations. The basic idea of the isomass three-body problem in terms of Hill’s equations is described below.
In the equimolar three-body problem, the masses of the three masses (A, B, and C) are assumed to be equal, and Hill’s equation describes the motion of the masses by dealing with minute changes in their relative positions and velocities.
In Hill’s equation, the position vectors of the three masses are expressed in terms of their relative positions as viewed from mass A. Specifically, the vector from A to B is x and the vector from A to C is y.
Hill’s equation is expressed as follows
d²x/dt² = -2ω dy/dt – 3ω² x d²y/dt² = 2ω dx/dt – 3ω² y
where ω is the angular velocity of the system and is expressed as ω² = Gm/(R³), where G is the universal gravitational constant, m is the mass of the mass point, and R is the average distance of the mass point.
Hill’s equation is a nonlinear differential equation, for which analytical solutions generally do not exist, making it difficult to solve analytically. Therefore, numerical analysis and approximate solution methods are generally used to solve Hill’s equation. Numerical methods include the Runge-Kutta method and the direct method (Direct method), which can be used to numerically predict the motion of the masses and understand the dynamics of the system.
Solving the Equivariant Three-Body Problem Using Poincaré Sections
The Poincaré section (Poincaré section) provides a method for observing the chaotic behavior of the isomaterial three-body problem. In the Poincaré section, the motion of a mass point is observed on a specific plane, and orbital intersections and characteristic patterns are analyzed to understand the dynamics of the system. The following is a basic procedure for solving the isomaterial three-body problem using the Poincaré cross section.
- Selection of the Poincaré cross section: In the Poincaré cross section, the plane in which the motion of a mass point is observed is selected. This plane is chosen based on the symmetry of the system and the property of interest, typically a plane with specific values of the relative positions and velocities of the mass points.
- Set initial conditions: Set initial conditions for the mass point on the selected Poincaré section. This includes values for the position and velocity of the mass, and the initial conditions are usually chosen according to the symmetry of the system and the property of interest.
- Calculation of the trajectory: Numerically calculate the motion of the mass point on the selected Poincaré cross section. For this, numerical integration methods (e.g. Runge-Kutta method) are used to predict the time variation of the position and velocity of the mass point. Here, starting from initial conditions, trajectories are computed at regular time intervals.
- Collecting data on a cross section: Data on the position and velocity of a pledge point are collected on a Poincaré cross section. Each time the calculated trajectory intersects the Poincaré cross section, the position and velocity of the masses at that point are recorded.
- Data analysis: The collected data is analyzed to examine the trajectory intersections and characteristic patterns. This includes analyzing characteristics such as the location of orbit intersections, periodic patterns, chaotic behavior, etc.
The use of Poincaré sections allows one to understand the chaotic behavior of the equimetric three-body problem. The observation of orbital intersections and characteristic patterns can also reveal the stability of the system, limits of predictability, and the existence of chaos. Furthermore, analysis using Poincaré cross sections can help visualize and visualize the results of numerical simulations.
Finally
Three-body problems are one of the remaining intractable problems in modern science, and various mathematical approaches to them have been utilized as algorithms for machine learning techniques.
Reference books for the three-body problem include “The Three-body Problem from Pythagoras to Hawking“
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