Joseph-Louis Lagrange1 was a French mathematician who analysed planetary motion. Newton's laws of motion and gravitation are very simple when you apply them to two bodies: a planet and a sun, for example. Once you introduce a third body, another planet, a moon or an asteroid, for example, they get much more complicated. Lagrange was one of the first people to study this problem, which is known as the 'three-body problem'. This 'three-body problem' has not been completely solved even today, but certain cases have been studied in detail.
One particularly interesting one is where one body is very large, one of middling size and one very small. This can be used to describe the motion of an asteroid, a planet and the Sun. Normally the planet orbits around the Sun in an elliptical orbit. The Sun is affected very slightly by this, but the Sun is so massive that it is hardly noticeable. In effect, the Sun remains fixed in the middle of the solar system. Lagrange discovered that there are five points called the Lagrange (or Lagrangian) points where the asteroid can share approximately the same orbit as the planet, with the same orbital period. As seen from the planet, the asteroid will appear to be fixed in the sky with respect to the Sun.
L1 is a point between the planet and the sun, but close to the planet. In the case of the Earth/Sun system, the L1 point is beyond the Moon's orbit, but still close to the Earth.
L2 is an equal distance from the planet as L1, but away from the sun.
L3 is a point in the orbit of the Earth, but on the far side of the Sun to the Earth.
L4 forms an equilateral triangle with the planet and the sun and orbits the sun at 60° ahead of the planet.
L5 forms an equilateral triangle with the planet and sun, and orbits the sun at 60° behind the planet.
A number of asteroids have been found to occupy Jupiter's L4 and L5 points, orbiting the Sun at the same distance as Jupiter, and 60° ahead and behind it. These asteroids bear the names of Trojan heroes, so the L4 point and the L5 point are often called 'Trojan points', with L4 holding the 'leading Trojans' and L5 the 'trailing Trojans'. Recently a few small asteroids have been found in the Trojan points of Mars, and there is an accumulation of dust at the Earth's Trojan points, but no asteroids.
The L1 point is unstable: small tugs from other gravitational bodies in the solar system will pull the object at the L1 point one way or another. As it leaves the L1 point, it can no longer orbit the Sun at the right speed, so it will start to drift off. For example, a spaceship in the Earth/Sun System's L1 point will wander off after about two weeks. But it is possible to keep a spacecraft at the L1 by using small amounts of rocket power. In fact the L1 is an ideal position for a telescope for observing the Sun, because there's nothing blocking the view and the telescope is always in the same place for transmitting back messages to Earth. There is in fact such a telescope sitting at the L1 point at the moment, called SOHO, the Solar and Heliospheric Observatory.
The L2 and L3 points are also unstable. Rumours of a supposed planet at the L3 point, where it would be invisible from the Earth, cannot be true, as the L3 point has a 'stability lifetime' of around 150 years.
The L4 and L5 are the only Lagrange points that are stable. Even then they are only stable if the Sun is more than 25 times the mass of the planet. A small object at the L4 point will stay there. If it is disturbed from the L4 point by a tug from another body, it will orbit around the L4 point (as seen from the planet). In actual fact, it is orbiting around the Sun in an elliptical orbit, but the period is such that with reference to the rotating frame based on the planet, it appears as a small rotation around L4. It is possible for a number of small bodies such as asteroids to occupy the L4 point, all apparently orbiting the point, as seen from the planet. Similarly, the L5 point is also stable.
Actual Occupied Lagrange Points
Every planet has a set of L4 and L5 points. As already mentioned, there are a number of asteroids in Jupiter's Trojan points. There are also Trojan points in the planet/moon systems, so that the Earth/Moon system has a pair of Trojan points on the Moon's orbit, 60° in front of and behind the Moon. There do not appear to be any small asteroids at these positions, but they appear to contain a cloud of dust too.
The Sun/Earth system's L1 point is occupied at the moment by a space observatory which is looking at the Sun. It gets an uninterrupted view of the sun. There are plans to put the NGST (Next Generation Space Telescope) at the L2 point. It will be far from Earth, so it will be free from light interference, but it will be in a fixed position relative to Earth, making it easy to track.
For more on Lagrange points, visit the Microwave Anisotropy Probe page.