How did the Apollo spacecraft use the Moon’s gravity to change their flight paths?

For the Apollo lunar missions, flight paths were calculated to utilize the Moon’s gravity to alter the spacecraft's trajectory. The spacecraft was placed on a path designed to pass ahead of the Moon, allowing lunar gravity to pull the craft and swing it around the far side.1

If no further maneuvers were performed, this gravitational interaction would have naturally looped the spacecraft around the Moon and returned it toward Earth. To enter lunar orbit, the astronauts performed a maneuver to reduce the spacecraft's flight speed while on the far side of the Moon, allowing the craft to be captured by lunar gravity.1

Navigating these paths involves the three-body problem of celestial mechanics, which accounts for the complex, mutual gravitational influences of Earth, the Moon, and the spacecraft. Because this problem has no general analytical solution, mission planners relied on high-speed digital computers to numerically integrate the equations of motion and precisely map the trajectory.1