The Lagrangian points (pron.: /l?rnd?i?n/; also Lagrange points, L-points, or libration points) are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be part of a constant-shape pattern with two larger objects (such as a satellite with respect to the Earth and Moon). The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit with them.
Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common center of mass, 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 matching the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined with the satellite's acceleration are in balance at the Lagrangian points, allowing the third body to be relatively stationary with respect to the first two bodies.
The three collinear Lagrange points (L1, L2, L3) were discovered by Leonhard Euler a few years before Lagrange discovered the remaining two.
In 1772, the Italian-French mathematician Joseph Louis Lagrange was working on the famous three-body problem when he discovered an interesting quirk in the results. Originally, he had set out to discover a way to easily calculate the gravitational interaction between arbitrary numbers of bodies in a system.
The logic behind this conclusion is that a system with one body is trivial, as it is merely static relative to itself; a system with two bodies is the relatively simple two-body problem, with the bodies orbiting around their common center of mass. However, once more than two bodies are introduced, the calculations become very complicated. It becomes necessary to calcu
ate the gravitational interaction between every pair of objects at every point along their trajectory.
Common opinion has falsely been that Lagrange himself considered how a third body of negligible mass would orbit around two larger bodies which were already in a near-circular orbit, and found that in a frame of reference that rotates with the larger bodies, there are five specific fixed points where the third body experiences zero net force as it follows the circular orbit of its host bodies (planets).
Actually, Lagrange considered in the first chapter of the Essai the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with conic section orbits. Thence, if one mass is made negligible, one immediately gets the five positions now known as the Lagrange Points; but Lagrange himself apparently did not note that.
In the more general case of elliptical orbits, there are no longer stationary points in the same sense: it becomes more of a Lagrangian “area”. The Lagrangian points constructed at each point in time, as in the circular case, form stationary elliptical orbits which are similar to the orbits of the massive bodies. This is due to Newton's second law (Force = Mass times Acceleration, or ), where p = mv (p the momentum, m the mass, and v the velocity) is invariant if force and position are scaled by the same factor. A body at a Lagrangian point orbits with the same period as the two massive bodies in the circular case, implying that it has the same ratio of gravitational force to radial distance as they do. This fact is independent of the circularity of the orbits, and it implies that the elliptical orbits traced by the Lagrangian points are solutions of the equation of motion of the third body.
Early in the 20th century, Trojan asteroids were discovered at the L4 and L5 Lagrange points of the Sun–Jupiter system.