When considering the Lagrange point 1 in the Sun/Earth system, Does Lagrange Point 1 have any orbital velocity around Earth? I suspect a body at L1 has no orbital velocity around earth. But consider the earth's position 6months later when it is opposite the sun. The Lagrange point would also have moved to the opposite side of the earth. This would suggest the orbital period of a body at L1 around the earth would be one year. So back to my original question: Does Lagrange Point 1 have any orbital velocity around Earth?