A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point
- First Online:
A new approach is developed here for resolving the Poisson equations in case the components of angular velocity of rigid body rotation can be considered as functions of the time parameter t only. A fundamental solution is presented by the analytical formulae in dependence on two time-dependent, real-valued coefficients. Such coefficients are proved to be the solutions of a mutual system of 2 Riccati ordinary differential equations (which has no analytical solution in the general case). All in all, the cases of analytical resolving of Poisson equation are quite rare (according to the cases of exact resolving of the aforementioned system of Riccati ODEs). So, the system of Euler–Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange’s case or (2) Kovalevskaya’s case or (3) Euler’s case or other well-known but particular cases (where the existence of particular solutions depends on the choice of the appropriate initial conditions).
Mathematics Subject Classification70E40 (integrable cases of motion)
Unable to display preview. Download preview PDF.
- 4.Synge J.L.: Classical dynamics. In: Flügge, S. (ed.) Handbuch der Physik, Principles of Classical Mechanics and Field Theory, vol. 3/1, Springer, Berlin (1960)Google Scholar
- 5.Ershkov S.V.: On the invariant motions of rigid body rotation over the fixed point, via Euler’s angles. Arch. Appl. Mech. 1–8 (2016, in press). http://link.springer.com/article/10.1007%2Fs00419-016-1144-6
- 8.Kamke, E.: Hand-book for Ordinary Differential Equations. Science, Moscow (1971)Google Scholar
- 10.Sanduleanu Sh.V., Petrov A.G.: Comment on New exact solution of Euler’s equations (rigid body dynamics) in the case of rotation over the fixed point. Arch. Appl. Mech. 1–3 (2016, in press). doi:10.1007/s00419-016-1173-1