How To Derive The L1 Lagrange Point

In the three-body problem there are 5 locations where a third object with negligible mass can be placed where gravitational forces of the two larger masses matches the centrifugal force required to orbit with the same period as secondary mass.

The L1, L2 and L3 Lagrange points all sit on a line that connects the primary and secondary masses. L1 is in-between the the two masses and must orbit with the same velocity as the secondary mass. Therefore, we need to work out at what distance along this line that will occur.

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