N-body problem of first-order differential equations, with initial conditions as initial position coordinates and initial momentum values. Symmetries in the -body problem yield global integrals of motion that simplify the problem. Translational symmetry of the problem results in the center of mass moving with constant velocity, so that , where is the linear velocity and is the initial position. The constants of motion and represent six integrals of the motion. Rotational symmetry results in the total angular momentum being constant where × is the cross product. The three components of the total angular momentum yield three more constants of the motion. The
Reduced mass a multitude of two-body problems, where classical mechanics is applicable. In a system with two point masses m and m such that they are co-linear, the center of mass may be found via formula_25. The moment of inertia of this system in the center of mass frame of reference is a two-body problem with two masses, each with its own distance to the center of mass. By converting to the equivalent reduced mass frame of reference, the system may be treated as a single reduced mass, μ, with a distance, d, equal to the sum of x and x. This