1. The three-body problem involves predicting the motion of three celestial bodies interacting gravitationally. 2. Newton's law of universal gravitation states that the force between two masses $m_1$ and $m_2$ separated by distance $r$ is given by $$F = G \frac{m_1 m_2}{r^2}$$ where $G$ is the gravitational constant. 3. For two bodies, Newton's equations can be solved exactly, giving predictable orbits like ellipses. 4. However, when three bodies interact, each body experiences forces from the other two, creating a system of coupled nonlinear differential equations. 5. These equations do not have a general closed-form solution, meaning we cannot write a simple formula to predict their exact future positions and velocities. 6. Small changes in initial conditions can lead to vastly different outcomes, a property known as chaos. 7. Therefore, Newton's equations can model the forces but cannot provide exact long-term predictions for three-body systems. 8. Instead, numerical methods and simulations are used to approximate their motions over time. 9. This unpredictability is why the three-body problem is fundamental in celestial mechanics and chaos theory.