Lagrange Points of the EarthMoon SystemA mechanical system with three objects, say the Earth, Moon and Sun, constitutes a threebody problem. The threebody problem is famous in both mathematics and physics circles, and mathematicians in the 1950s finally managed an elegant proof that it is impossible to solve. However, approximate solutions can be very useful, particularly when the masses of the three objects differ greatly. For the SunEarthMoon system, the Sun's mass is so dominant that it can be treated as a fixed object and the EarthMoon system treated as a twobody system from the point of view of a reference frame orbiting the Sun with that system. 18th century mathematicians Leonhard Euler and JosephLouis Lagrange discovered that there were five special points in this rotating reference frame where a gravitational equilibrium could be maintained. That is, an object placed at any one of these five points in the rotating frame would stay there, with the effective forces with respect to this frame canceling. Such an object would then orbit the Sun, maintaining the same relative position with respect to the EarthMoon system. These five points were named Lagrange points and numbered from L1 to L5.
The L5 point was the focus of a major proposal for a colony in "The High Frontier" by Gerard K. O'Neill and a major effort was made in the 1970's to work out the engineering details for creating such a colony. There was an active "L5 Society" that promoted the ideas of O'Neill. The L4 and L5 points make equilateral triangles with the Earth and Moon. The Lagrange points L1, L2 and L3 would not appear to be so useful because they are unstable equilibrium points. Like balancing a pencil on its point, keeping a satellite there is theoretically possible, but any perturbing influence will drive it out of equilibrium. However, in practice these Lagrange points have proven to be very useful indeed since a spacecraft can be made to execute a small orbit about one of these Lagrange points with a very small expenditure of energy. They have provided useful places to "park" a spacecraft for observations. These orbits around L1 and L2 are often called "halo orbits". L3 is on the opposite side of the Earth from the Moon, so is not so easy to use.

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ThreeBody Equipotential SurfacesA mechanical system with three objects, say the Earth, Moon and Sun, constitutes a threebody problem. The threebody problem is famous in both mathematics and physics circles, and mathematicians in the 1950s finally managed an elegant proof that it is impossible to solve. However, approximate solutions can be very useful, particularly when the masses of the three objects differ greatly. One of the contributions of Lagrange was to plot contours of equal gravitational potential energy for systems where the third mass was very small compared to the other two. Below is a sketch of such equipotential contours for a system like the EarthMoon system. The equipotential contour that makes a figure8 around both masses is important in assessing scenarios were one partner loses mass to the other. These equipotential loops form the basis for the concept of the Roche lobe. Contours of Equal Gravitational PotentialOne of Lagrange's observations from the potential contours was that there were five points at which the third body could be at equilibrium, points which are now referred to as Lagrange points. The Lagrange Points for a system like the EarthMoon systemThe Lagrange points L_{1}, L_{2}, and L_{3} are unstable equilibrium points. Like standing a pencil on its point, it is possible to achieve equilbrium, but any displacement away from that equilibrium would lead to forces that take it further away from equilibrium. Remarkably, the Lagrange points L_{4} and L_{5} are stable equilibrium points for the small mass in the threebody system and this threebody geometry could be maintained as M_{2} orbited about M_{1}.

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Joseph Louis LagrangeLagrange was and 18th century mathematician who tackled the famous "threebody problem" in the late 1700s. The problem cannot be solved exactly, but he found that in the case where the third body is very small compared to the other two, some useful approximate solutions could be found. 
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