# Definitions of Lagrange points: $L_4$ and $L_5$

We have the the five Lagrange points (let consider Earth and Sun):

• $L_1$ - lie between Sun and Earth;
• $L_2$ - beyond the Earth;
• $L_3$ - beyond the Sun;

And what's the difference between $L_4$ and $L_5$? Does they defined with the respect to rotation of Earth around the Sun?

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Closely related (not an exact duplicate) to physics.stackexchange.com/q/36092 – Waffle's Crazy Peanut Dec 5 '12 at 14:12

The $L_4$ and $L_5$ points are positioned such that they are an equal distance away from both bodies (the Earth and the Sun in this case). They are chosen such that $L_4$ lies ahead of the smaller body in it's orbital path while the $L_5$ point is behind the smaller body in it's orbital path. See this article for more details.

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L4 and L5 points are the most stable for a given orbiting celestial body. The only difference is that L4 lies in front of the body while L5 is ahead of it as @tpg quotes it. They are almost the same because both form an equilateral triangle with the bodies. If you sketch both, you'd get a 60 degree angle. These Lagrange points are handled with care while dealing with celestial mechanics and it was the mysterious (completely math-based) solution for our miracled  three-body problem. That's all...

Does they defined with the respect to rotation of Earth around the Sun?

Their definition swirls around in orbital motion of a body. A body that orbits the sun along with Earth could be placed at L4 or L5. The mass of the body could be larger, indicating the stability of these points. On the other hand, L1, L2 and L3 are almost unstable (only negligible masses could stay in them) and even a slight perturbation could cause a great deviation in the configuration of orbiting body.

So the main conclusion of this equilateral triangle is - Gravitational equilibrium. The masses of other two bodies balance each other, thereby equilibrating the third (orbiting) body.

You might be interested in this one already here - Why are L4 and L5 lagrangian points stable? and a good podcast about these points. Greedy for more math..?

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