Abstract: For two mechanical contest problems, two simplified coordinate systems are used, that is, the "Serret--Andoyer approximate coordinate system" and the "virtual coordinate system" , respectively, based on the Euler dynamics equation. On this basis, the solutions of the stability conditions of the small ball in the synchronous rotation of the big ball in the 9th problem of the 1st national youth mechanics competition are is discussed. The results show that it has lower stability critical speed requirements. This will be more conducive to improving the stability of the rotation structure. For the problem of rotation balls in the 3rd problem of the 12th National Zhou Peiyuan University Mechanics Competition, the universal motion law at any latitude is derived. When the initial angular momentum of the system is zero, a simple and magical motion mode occurs.