The three-body problem, and more generally the many-body problem, has a wide range of applications in science. It can be found in statistical and quantum physics, fluid mechanics and celestial mechanics. Understanding how electrons in metals or semiconductors interact with each other and with the atoms of the crystal lattice allows us to design new materials with useful electrical, magnetic or mechanical properties. But let’s start at the beginning! 

The three-body problem is a classic problem in mechanics. It refers to a situation in which three objects interact with each other, for example through gravity. It is also one of the simplest cases of so-called deterministic chaos. (1)

Deterministic chaos is a term used to describe systems whose behaviour appears to be completely chaotic, despite the fact that the behaviour of each of the elements can be accurately described. It is also manifested by the fact that a minimal change in the initial conditions can have a huge effect on the behaviour of the system. (2) Although there are counterexamples, the apparent chaos is usually due to the fact that the equations and relationships describing the system are very complex. So complex, in fact, that it is often not even possible to obtain an exact analytical solution, but only a numerical approximation. (3) This is where the unusual nature of the three-body problem comes in. Although we are able to solve many such systems to a high degree of accuracy, such a seemingly simple system already manifests chaotic tendencies, and in many cases the solution is completely impossible without advanced numerical methods. (4) Systems of three or more bodies may or may not behave chaotically [1]. A good example is the solar system. It is made up of a huge number of elements, each interacting with the others, and a complete analytical solution is absolutely inconceivable! However, using some approximations, we can calculate the trajectories of many objects very accurately. (5)

Let us consider the configuration of the Sun-Earth-Moon system. This is a model three-body system, but there is very little chaos in it. This is due to the large differences in the masses of the three objects. This allows us to think of this system as the Earth orbiting the Sun and the Moon orbiting the Earth. This is a good approximation because it can be shown that the influence of other objects in the solar system does not permanently change the orbit of either the Earth or the Moon. A similar approximation is true for the other planets in the Solar System, and is what distinguishes the planets from the dwarf planets. In a sense, Pluto has become a victim of the three-body problem. (6)

The motion of the planets is an example of a many-body system without chaos. However, we can find other objects in the solar system such as Zoozve, the quasi-moon of Venus. This is an object that moves in the gravitational field of Venus, but is far enough away from Venus that it is also influenced by the Earth, Mercury and the Sun. This causes Zoozve’s orbit to change greatly over time, and it will leave Venus in about 500 years. (7)

Some facts about the many-body problem:

• Non-linearity of the equations: The three-body equations of motion are non-linear, which means that the relationships between the variables are not simple ratios, making it much more difficult to solve a system of such equations. They cannot be summed or simplified in the same way as linear equations. Non-linearity also means that small changes in the initial conditions can cause very large differences in the behaviour of the system, which is a characteristic of chaotic systems.
• Lack of a general closed-form solution: While simple solutions exist for the two-body problem (such as Kepler’s laws for planetary motion), such solutions are not always possible for the three-body problem. Henri Poincaré, a French mathematician, was the first to prove at the turn of the 20th century [2] that there is no single, general analytical solution to the three-body problem in closed form, i.e. consisting of a finite sequence of basic operations and functions. This means, among other things, that three bodies rarely move along nice curves, such as an ellipse. Poincaré’s work revealed the fundamental complexity of the problem and led to the development of chaos theory. It was not long before a general solution was found, but it turned out to be terribly inconvenient to compute (Sundman’s theorem)[1].
• Practical applications: The three-body problem is central to the design of space missions. For example, engineers use libration points – specific points in space where a satellite can be placed so that its orbit is stable. Understanding the interactions within three and more bodies allows space trajectories to be planned with minimal fuel consumption, and enables travel to much more distant corners of the solar system.
• Numerical Solutions and Simulations: In the computer age, scientists can use advanced numerical techniques to simulate trajectories in many-body problems. These simulations are essential in many areas of science and engineering, from predicting the motion of asteroids that could threaten Earth to modelling the evolution of entire galaxies[2].

The problem of chaos is one of the areas where physics is being reinvented. The 2021 Nobel Prize in Physics was awarded for “pioneering contributions to the understanding of complex physical systems”. Scientists were honoured for research that, among other things, helps model the behaviour of global warming (meteorology is an excellent example of chaos). This research has also contributed to the understanding of many other phenomena: from the behaviour of atoms, brain activity, flocks of birds, the evolution of glaciers and planetary motion.

As we can see, the three-body problem is very important in physics in general. However, it turns out that much less obvious situations can be considered in a similar way, and applications of the many-body problem can be found in many fields:

Economics

In economics, the many-body problem refers to the interaction between different market agents such as consumers, firms, banks and governments. Each of these agents influences the others through their economic decisions, just as celestial bodies influence each other through the force of gravity.

For example, models of financial markets often assume that asset prices are the result of interactions between different market participants, each with their own strategies and information. Small changes in the behaviour of one participant can lead to large and unexpected changes in the whole system, similar to the chaotic behaviour in the three-body problem.

Biology

In biology, the many-body problem manifests itself in the interactions between different organisms in an ecosystem and also, for example, in gene expression [3]. Such models have been used to predict the development of pandemics, including COVID-19. In both cases, understanding these dynamic, complex systems requires the use of sophisticated mathematical and numerical simulation methods to predict the behaviour of the systems based on inputs and initial conditions, as in celestial mechanics. Although these analogies are simplistic, they allow a better understanding of how complex interactions affect whole systems, which is crucial in both economics and biology. Another example is biological evolution. “Chance and necessity”, as Jaques Monod put it, are the key phenomena that led to the emergence of complex biological organisms. 

Chemistry

In chemistry, the many-body problem manifests itself in molecular dynamics, where computer simulations are used to model the motions and interactions of atoms and molecules in processes such as chemical reactions, protein folding or ligand binding. These simulations help to understand mechanisms at the molecular level, which is extremely important in the design of drugs and new materials.

Computing

In computer science, numerical methods developed for solving many-body problems are used in optimisation algorithms and simulations. For example, algorithms used to find the shortest path in logistics problems or to simulate the behaviour of crowds use similar approaches to those used to solve multi-agent interaction problems.

The humanities

The three-body problem and the notion of complex dynamical systems, although originally developed in the context of physics, have also found application in the humanities. These disciplines are not concerned with the literal application of mathematical models of the three-body problem, but with the metaphorical application and conceptual understanding of complex systems that are fundamental to understanding human behaviour and social interaction. 

In sociology, the concept of complex systems is used to analyse and model social interactions, social structures and group dynamics. For example, modelling complex social networks can help to understand how information or behaviour spreads through society. Figuratively speaking, the three-body problem can be used to study how the introduction of a new element (e.g. a technology, idea or policy) can affect existing social relationships and power structures.

Historians, on the other hand, can study how different forces (economic, social, political) interact to produce unexpected outcomes of historical events, as in the three-body problem, where the trajectories of the bodies are unpredictable when they interact. Unfortunately, the great complexity of the human world can only be understood retrospectively, and even if we know the present state of affairs, we cannot predict what is to come. As the Danish physicist Niels Bohr summed it up: “Prediction is very difficult, especially if it’s about the future!”.

Photo: Unsplash

Bibliography:

[1]  Z E Musielak, B Quarles 2014 Rep. Prog. Phys. 77 065901 https://iopscience.iop.org/article/10.1088/0034-4885/77/6/065901
[2] The Three-Body Problem, Richard Montgomery https://www.scientificamerican.com/article/the-three-body-problem/
[3] Masaki Sasai, Peter G. Wolynes, Stochastic Gene Expression as a Many Body Problem  https://arxiv.org/ftp/cond-mat/papers/0301/0301365.pdf