On certain two-dimensional conservative mechanical systems with a cubic second integral

In a previous paper (Yehia H M 1986 J. Mec. Theor. Appl. 5 55–71) we have introduced a method for constructing integrable conservative two-dimensional mechanical systems whose second integral of motion is polynomial in the velocities. This method has proved successful in constructing a multitude of irreversible systems (involving gyroscopic forces) with a second quadratic integral (Yehia H M 1992 J. Phys. A: Math. Gen. 25 197–221). The objective of this paper is to apply the same method for the systematic construction of mechanical systems with a cubic integral. As in our previous works, the configuration space is not assumed to be a Euclidean plane. This widens the range of applicability of the results to diverse mechanical systems to include such problems as rigid body dynamics. Several new reversible and irreversible integrable systems are obtained. Some of these systems generalize previously known ones by introducing additional parameters which may change either or both of the configuration manifold and the potential of the forces acting on the system. Other systems are completely new. An application is given to problems of rigid body dynamics. The famous classical integrable case due to Goriachev and Chaplygin and all its subsequent generalizations by several authors are further generalized to include certain variable gyroscopic forces that preserve a cubic integral. On the other hand, the above case and another less famous case of rigid body dynamics due to Goriachev and the Hall–Toda case of particle mechanics are all obtained as special cases, corresponding to different choices of certain parameters, from one more general system unifying them all.