Mathematical equations can predict the motions of two gravitating masses. But when a third or more are introduced, our analytical tools fall short. It is impossible to write down all the terms of a general formula that can accurately describe the motion of three or more gravitating objects. The problem is in knowing how many unknown variables an n-body system contains.

The three-body problem

Thanks to Isaac Newton, we can write a set of equations to describe the gravitational force that acts between bodies. But, when we try to find a general solution for the unknown variables in these equations, we are faced with a mathematical limitation.For every unknown, there must be one equation that describes it. A two-body system is solvable because its gravitational attraction influences the paths they take in such a way that a simple mathematical formula can describe it. But three or more orbiting objects leaves us with more variables than equations that can describe them.

The mathematical constraint

A system of three stars could come crashing or flung out of orbit after a long time of apparent stability. Almost every possible configuration is unpredictable on long timescales. A slight difference in position and velocity could have a large range of potential outcomes. Physicists describe this behaviour as chaotic - an important characteristic of n-body systems. But it is still deterministic; it’s not random.

Objects that move according to analytically unsolvable equations of motion

Advancements in computer simulations help to avoid disaster. By approximating solutions with powerful processors, we can better predict the motion of n-body systems on long time scales.   When one body in a group of three is so light it has no significant force on the other two, the system behaves as a two-body system. This is known as the “restricted three-body problem.” The restricted three-body system is useful to describe an asteroid in the Earth-Sun gravitational field or a small planet in the field of a black hole and a star.

The restricted three-body system

For space missions to avoid catastrophe, complicated orbits must be calculated with great precision. 

Download a free audiobook version of "The Three-Body Problem" and support TED-Ed's nonprofit mission: https://www.audible.com/ted-ed -- In 2009, researchers ran a simple experiment. They took everything we know about our solar system and calculated where every planet would be up to 5 billion years in the future. They ran over 2,000 simulations, and the astonishing variety in results revealed that our solar system may be much less stable than it seems. Fabio Pacucci explores the n-body problem and the motion of gravitating objects. Lesson by Fabio Pacucci, directed by Hype CG. Animator's website: https://www.hype.cg/ and https://www.luisacopetti.com/ Sign up for our newsletter: http://bit.ly/TEDEdNewsletter Support us on Patreon: http://bit.ly/TEDEdPatreon Follow us on Facebook: http://bit.ly/TEDEdFacebook Find us on Twitter: http://bit.ly/TEDEdTwitter Peep us on Instagram: http://bit.ly/TEDEdInstagram View full lesson: https://ed.ted.com/lessons/newton-s-three-body-problem-explained-fabio-pacucci Thank you so much to our patrons for your support! Without you this video would not be possible! João Henrique Rodrigues, Sebastiaan Hols, Aries SW, SANG HAN, Amy Lopez, ReuniteKorea, Vinh-Thuy Nguyen, Liz Candee, Clovis Norroy, Danielle Downs, Nik Maier, Angel Pantoja, Nishant Suneja, 张晓雨, Srinivasa C Pasumarthi, Kathryn Vacha, Anthony Arcis, Jeffrey Segrest, Sandra Fuller Bocko, Alex Pierce, Lawrence Teh Swee Kiang, BRENDAN NEALE, Jane White, Karmi Nguyen, John C. Vesey, Yelena Baykova, Harshita Jagdish Sahijwani, Won Jang, Nick Johnson, Tariq Keblaoui, Carlos H. Costa, Eimann P. Evarola, Aleksandr Lyozin, Mohamed Elsayed, Alan Wilder, Marcus Appelbaum, Francisco Leos, Kevin O'Leary, Les Howard, Ten Cha, Mehmet Yusuf Ertekin, Arlene Weston, phkphk123321, Jennifer Kurkoski, Ryan B Harvey, Austin Randall, Abhishek Bansal, Jayant Sahewal, Dian Atamyanov and igor romanenko.

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