Inner Lagrange Point - In Lockstep around the Sun

The Earth moves gravitationally bound around the Sun within a year. Any mass that orbits the Sun at a greater distance takes longer for one orbit, while objects closer to the central star take less time. This simple relationship generally applies in a two-body system. However, when a third mass is involved we are dealing with a three-body problem. If we then wanted to determine the orbits of these three bodies, we would not find that an analytical solution.