page_title: Lagrange point
section_title: Mathematical details
hierarchical_section_title: Lagrange point / Mathematical details
context_page_description: In celestial mechanics, the Lagrange points are orbital points near two large co-orbiting bodies. At the Lagrange points the gravitational forces of the two large bodies cancel out in such a way that a small object placed in orbit there is in equilibrium relative to the center of mass of the large bodies.
  There are five such points, labeled L₁ to L₅, all in the orbital plane of the two large bodies. L₁, L₂, and L₃ are on the line through the centers of the two large bodies, while L₄ and L₅ each act as the third vertex of an equilateral triangle formed with the centers of the two large bodies. L₁, L₂, L₃ are unstable equilibria, whereas L₄ and L₅ are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.
  For each given combination of co-orbiting planetary bodies there are five Lagrange points L₁ to L₅ for the Sun–Earth system, and in a similar way there are five different Lagrange points for the Earth–Moon system. Several planets have trojan satellites near their L₄ and L₅ points with respect to the Sun. Jupiter has more than a million of these trojans.
context_section_description: Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common barycenter, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.
caption: A contour plot of the effective potential due to gravity and the centrifugal force of a two-body system in a rotating frame of reference. The arrows indicate the gradients of the potential around the five Lagrange points—downhill toward them (red) or away from them (blue). Counterintuitively, the L4 and L5 points are the high points of the potential. At the points themselves these forces are balanced.