What is a Lagrange Point?
Credit  NASA/WMAP Science Team 

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What are they?: Lagrange Points
Lagrange points are positions in space where objects sent there tend to stay put. At Lagrange points, the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. These points in space can be used by spacecraft to reduce fuel consumption needed to remain in position.
Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. These can be used by spacecraft to reduce fuel consumption needed to remain in position.
Lagrange points are named in honor of ItalianFrench mathematician JosephyLouis Lagrange.
There are five special points where a small mass can orbit in a constant pattern with two larger masses. The Lagrange Points are positions where the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. This mathematical problem, known as the "General ThreeBody Problem" was considered by Lagrange in his prize winning paper (Essai sur le Problème des Trois Corps, 1772).
Of the five Lagrange points, three are unstable and two are stable. The unstable Lagrange points  labeled L1, L2 and L3  lie along the line connecting the two large masses. The stable Lagrange points  labeled L4 and L5  form the apex of two equilateral triangles that have the large masses at their vertices. L4 leads the orbit of earth and L5 follows.
The L1 point of the EarthSun system affords an uninterrupted view of the sun and is currently home to the Solar and Heliospheric Observatory Satellite SOHO.
The L2 point of the EarthSun system was the home to the WMAP spacecraft, current home of Planck, and future home of the James Webb Space Telescope. L2 is ideal for astronomy because a spacecraft is close enough to readily communicate with Earth, can keep Sun, Earth and Moon behind the spacecraft for solar power and (with appropriate shielding) provides a clear view of deep space for our telescopes. The L1 and L2 points are unstable on a time scale of approximately 23 days, which requires satellites orbiting these positions to undergo regular course and altitude corrections.
NASA is unlikely to find any use for the L3 point since it remains hidden behind the Sun at all times. The idea of a hidden planet has been a popular topic in science fiction writing.
The L4 and L5 points are home to stable orbits so long as the mass ratio between the two large masses exceeds 24.96. This condition is satisfied for both the EarthSun and EarthMoon systems, and for many other pairs of bodies in the solar system. Objects found orbiting at the L4 and L5 points are often called Trojans after the three large asteroids Agamemnon, Achilles and Hector that orbit in the L4 and L5 points of the JupiterSun system. (According to Homer, Hector was the Trojan champion slain by Achilles during King Agamemnon's siege of Troy). There are hundreds of Trojan Asteroids in the solar system. Most orbit with Jupiter, but others orbit with Mars. In addition, several of Saturn's moons have Trojan companions.
In 1956 the Polish astronomer Kordylewski discovered large concentrations of dust at the Trojan points of the EarthMoon system. The DIRBE instrument on the COBE satellite confirmed earlier IRAS observations of a dust ring following the Earth's orbit around the Sun. The existence of this ring is closely related to the Trojan points, but the story is complicated by the effects of radiation pressure on the dust grains.
In 2010 NASA's WISE telescope finally confirmed the first Trojan asteroid (2010 TK7) around Earth's leading Lagrange point.
Finding the Lagrange Points
The easiest way to understand Lagrange points is to think of them in much the same way that wind speeds can be inferred from a weather map. The forces are strongest when the contours of the effective potential are closest together and weakest when the contours are far apart.
This page was originally written (with mathematical equations) by Neil J. Cornish of the Wikinson Microwave Anistropy Probe team.