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).
This is a preview of subscription content,to check access.
Access this article
Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn. Pergamon Press, New York (1976)
Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Boston (1980)
Symon, K.R.: Mechanics, 3rd edn. Addison-Wesley, Boston (1971)
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)
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
Gashenenko, I.N., Gorr, G.V., Kovalev, A.M.: Classical Problems of the Rigid Body Dynamics. Naukova Dumka, Kiev (2012)
Llibre, J., Ramírez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73(4), 2273–2290 (2013)
Kamke, E.: Hand-book for Ordinary Differential Equations. Science, Moscow (1971)
Ershkov, S.V.: A procedure for the construction of non-stationary Riccati-type flows for incompressible 3D Navier–Stokes equations. Rend. Circolo Mat. Palermo 65(1), 73–85 (2016)
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
Popov, S.I.: On the motion of a heavy rigid body about a fixed point. Acta Mech. 85(1), 1–11 (1990)
Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226(8), 2461–2472 (2015)
Ismail, A.I., Amer, T.S.: The fast spinning motion of a rigid body in the presence of a gyrostatic momentum \(l3\). Acta Mech. 154(1), 31–46 (2002)
I am thankful to Dr. Hamad H. Yehya for the insightful motivation during the fruitful discussions in the process of preparing of this manuscript.
About this article
Cite this article
Ershkov, S.V. A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point. Acta Mech 228, 2719–2723 (2017). https://doi.org/10.1007/s00707-017-1852-1