New solutions of classical problems in rigid body dynamics

• Integrable cases are very rare in rigid body dynamics, and so are particular solutions of equations of motion.
• The paper introduces a new particular solution in each of the two classical problems of motion.
• (a) A rigid body about a fixed point in a Newtonian gravitational field.
• (b) A free rigid body in a liquid medium.

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.