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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 8958747, 10 pages
Research Article

Integrable 2D Time-Irreversible Systems with a Cubic Second Integral

1Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2Faculty of Science, Mathematics Department, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received 20 September 2015; Accepted 14 February 2016

Academic Editor: Yao-Zhong Zhang

Copyright © 2016 H. M. Yehia and A. A. Elmandouh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces. Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev.