New integrable problems in the dynamics of particle and rigid body

Abstract

In the present article, a new two-dimensional integrable system containing 17 free parameters is introduced. For giving certain values for these parameters, new integrable problems can be constructed, which generalize some known previous problems, and in some cases, we can restore some previous integrable problems. Two new integrable problems are announced, describing the motion in an Euclidean plane and on a pseudo-sphere. In the irreversible case, a new integrable problem in rigid body dynamics, which generalizes Goriachev–Chaplygin’s case (Varshav Univ Izvest 3:1–13, 1916), Yehia’s case (Mech Res Commun 23:423–427, 1996) and Elmandouh’s case (Acta Mech 226:2461–2472, 2015), is announced.