Written by Kat Jivkova
In 1937, renowned English mathematician Edmund Whittaker described the three body problem as “the most celebrated of all dynamical problems”. Its criteria are as follows: three gravitating bodies move in space, in relation to one another; given their condition, determine their subsequent motion. Ostensibly, this statement seems simplistic, rendering the solution to the problem suspiciously straightforward. In reality, the three body problem perplexed the vast majority of mathematicians operating in the eighteenth century, largely because of the mathematical complexities that surrounded its solution. While Newton’s theory of gravity, formulated in the late seventeenth century, sufficiently explained the interactions between two particles and their mutual gravitational attraction, the question of how to apply this to more than two bodies remained challenging – the former could be solved through simple elementary functions, whereas the latter existed as a complicated nonlinear problem. Historical scholarship centred on the three body problem vis-à-vis the rise of celestial mechanics in the eighteenth century remains scant at best. To my knowledge, the only work which comprehensively addresses this branch of physics in recent years has been Costantino Sigismondi and Paolo De Vincenzi’s work on solar and lunar eclipses, from early antiquity to the present. I re-situate the three body problem inside their research in order to shed light on its significance in the development of gravity law in the era of Newton.
Indeed, Isaac Newton was the first physicist to raise the question of how two masses could move in space, provided that the only force exerted onto them was their mutual gravitational attraction (N=2). In his seminal work published in 1687, Principia, he solves this problem by identifying all of the possible orbits that his two bodies could hypothetically perform. However, he discovered that it was far more difficult to find periodic orbits of three body systems (N=3). In fact, no mathematician was able to find a single one of these orbits in the next two centuries in closed form. By the late nineteenth century, the French mathematician and theoretical physician Henri Poincare was finally able to pinpoint exactly why this was the case. For one, the trajectories of three body systems were considerably less predictable to those of two, meaning that it was almost impossible to accurately trace them. Poincare went as far as to claim that most of these trajectories were not periodic at all, bar a few. Further, most of the mathematicians that were concerned with identifying the few trajectories that did exist operated in a period in which electronic computers did not. This complicated the numerical search for such orbits even further. Even with the existence of computers in the present, the three body problem remains an open-ended question that will probably never be fully realised.
As mentioned, the three body problem became integrated into the sub-branch of theoretical physics, celestial mechanics, during the eighteenth century. In simple terms, celestial mechanics is concerned with the motions of objects in outer space. Hence, the three body problem became a critical phenomenon in the understanding of how multiple celestial bodies moved in space and interacted with one another. Aside from Newton, other mathematicians embarked on similar missions to uncover the three body problem. During the eighteenth century, the popularisation of celestial mechanics meant that mathematics was no longer constrained to the laws of geometry. Alexis Clairaut, for instance, successfully published his first approximate solution to the problem in 1747. By looking at the problem outside of elementary function, he was able to reach an approximation that closely accounted for the three body motion – this discovery granted him the St. Petersburg Academy prize five years later. Leonard Euler, on the other hand, identified three particular solutions to the general problem two decades later, for which he was awarded the Prix de l’Academie de Paris alongside his colleague, Joseph-Louis Lagrange. Meanwhile, Lagrange himself was responsible for the discovery of two families of orbits.
By the late eighteenth century, it was becoming increasingly evident that the three body problem could not be solved in closed form. Certainly, mathematicians reluctantly accepted that its solutions could never reach a level of detail remotely comparable to Newton’s two body problem. After all, once a third element was added to a previously stabilised system of two objects, chaos was quite literally unleashed. Subsequently, the mathematician and physicists working on the problem diverted their attention to improving the approximations of their solutions, using finite segments to predict periodic orbits. However, these efforts failed to generate any further formulas for the orbits, which still applies to the present day. In spite of these unfortunate circumstances, the three body problem provides us with a lens through which we can view developments in celestial mechanics after Newton’s Principia.
Ideas relating to the motion of planets in our solar system, which were a central component to the three body problem, guided the development of celestial mechanics in the late eighteenth century. On a practical level, the problem manifested in ideas of Earth-Moon-Sun relative motion, with the Earth and Sun representing the components of Newton’s original “two body” system, and the Moon acting as the third, less-big, element of the newly-imagined three body system. This raised the question of how these three entities balanced each other, and whether there would come a time during which the Moon could crash into the Earth. Another prime concern was the relationship between the Earth, Sun and Venus, which orbited on the inside lane below Earth – mathematicians even speculated that Venus could potentially throw Earth off its regular orbit, causing it to sink into the Sun’s rays. There was a genuine concern among mathematicians that, until the three body problem was solved, mankind could never know the fate of their planet for certain. Notably, the motion of the Moon, or Venus, in the combined gravitational field of the Sun and Earth remains a conundrum among physicists of the twenty-first century.
Newton’s solution to the two body problem, and the introduction of the three body problem to the field of celestial mechanics can be seen by mathematicians as both a blessing and a curse. On the one hand, the three body problem promises to explain how three gravitational objects can be stabilised and predicts what their orbits might look like. This can be applied on a practical level to look at the relationship between the Earth, Moon and Sun, for instance. On the other hand, to the frustration of seventeenth-century mathematicians (and probably present day ones too), the problem is technically unsolvable. In other words, there is not one particular solution that can be applied to every single three objects and their motion. The complicated interplay of gravitational forces that accompanies the three body problem means that this issue will likely remain a mystery to scholars of celestial mechanics, at least for the time being.
Bibliography
Barrow-Green, June. Poincaré and the Three Body Problem. Providence, R.I: American Mathematical Society, 1997.
Charpentier, Éric, E. Ghys, and Annick Lesne, eds. The Scientific Legacy of Poincaré. Translated by Joshua Bowman. Providence, Rhode Island: American Mathematical Society, 2010.
Liao, Shijun, Xiaoming Li, and Yu Yang. “Three-Body Problem: From Newton to Supercomputer plus Machine Learning.” New astronomy 96 (2022): 1-11.
Newton, Isaac, I. Bernard Cohen, and Anne Whitman. The Principia Mathematical Principles of Natural Philosophy. A New Translation by I. Bernard Cohen and Anne Whitman; Assisted by Julia Budenz; Preceded by a Guide to Newton’s Principia by I. Bernard Cohen. Berkeley: University of California Press, 1999.
Sigismondi, Costantino, and Paolo De Vincenzi. “Eclipses: A Brief History of Celestial Mechanics, Astrometry and Astrophysics.” Universe (Basel) 10, no. 2 (2024): 90-100.
Valtonen, Mauri., Joanna. Anosova, Konstantin. Kholshevnikov, Aleksandr. Mylläri, Victor. Orlov, and Kiyotaka. Tanikawa. The Three-Body Problem from Pythagoras to Hawking. Cham: Springer International Publishing, 2016.
Featured Image Text: Isaac Newton, Sir Isaac Newton’s Own First Edition Copy of His Philosophiae Naturalis Principia Mathematica with His Handwritten Corrections for the Second Edition., accessed April 7, 2024, https://commons.wikimedia.org/wiki/File:NewtonsPrincipia.jpg.
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