The L4 and L5 Lagrange points, 60 degrees in front of and behind the Moon in its orbit around Earth, orbit at the same speed as the Moon. But when a spacecraft flies to the Moon, the gravity of the Moon captures it, and then it only has to expend fuel to set up its orbit around the Moon the way it wants.
Though the gravity of the Earth and the Moon balance at the L4 and L5 points, there is nothing at those points to pull a spacecraft into orbit around them. So, does a vessel have to brake to set up an orbit around those points? How does the delta V to orbit those points compare to going from Earth into orbit around the Moon?
Image courtesy NASA, originally posted here