Lagrange points


January 24, 2022

Lagrange points, libration points (lat. librātiō - rocking) or L-points - points in a system of two massive bodies, in which a third body with a negligible mass, which is not affected by any other forces, except for gravitational forces from the first two bodies, can remain motionless relative to these bodies. More precisely, Lagrange points are a special case in solving the so-called restricted three-body problem - when the orbits of all bodies are circular and the mass of one of them is much less than the mass of any of the other two. In this case, we can assume that two massive bodies revolve around their common center of mass with a constant angular velocity. There are five points in the space around them where a third body with a negligible mass can remain motionless in the rotating frame of reference associated with massive bodies. At these points, the gravitational forces acting on the small body are balanced by the centrifugal force. Lagrange points got their name in honor of the mathematician Joseph Louis Lagrange, who was the first to give a solution of a mathematical problem in 1772, from which the existence of these singular points followed.

Location of Lagrange points

All Lagrange points lie in the plane of orbits of massive bodies and are denoted by the capital Latin letter L with a numerical index from 1 to 5. The first three points are located on a line passing through both massive bodies. These Lagrange points are called collinear and are denoted L1, L2 and L3. Points L4 and L5 are called triangular or Trojan. Points L1, L2, L3 are points of unstable equilibrium, at points L4 and L5 the equilibrium is stable. L1 is located between two bodies of the system, closer to the less massive body; L2 - outside, behind a less massive body; and L3 - for the more massive. In a coordinate system with the origin at the center of mass of the system and with an axis directed from the center of mass to a less massive body, the coordinates of these points in the first approximation in α are calculated using the following formulas: r one ( R [ one − ( α 3

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