Absolute and relative choreographies in rigid body dynamics

For the classical problem of motion of a rigid body about a fixed point with zero integral of areas, the paper presents a family of solutions which are periodic in the absolute space. Such solutions are known as choreographies. The family includes the famous Delaunay solution in the case of Kovalevskaya, some particular solutions in the Goryachev-Chaplygin case and Steklov’s solution. The «genealogy» of the solutions of the family, arising naturally from the energy continuation, and their connection with the Staude rotations are considered.

It is shown that if the integral of areas is zero, the solutions are periodic but with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).