Singularities in Classical Mechanical Systems

Singularities in the equations of motion of a classical mechanical system usually play a dominant role in the global phase portrait of the system. By a singularity we mean a point or set of points where the system is undefined, as in the case of a collision between two or more of the particles in the n-body problem. Such singularities often lead to a complicated global orbit structure. Not only do certain solutions tend to run off the phase space, but also nearby solutions tend to behave in an erratic or unpredictable manner. Numerical studies of such systems are often inconclusive because of this erratic behavior. And power series or other analytic techniques often yield only a very local description of solutions near the singularity, one which gives no hint of the global complexity of the system.