A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point

Abstract

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).

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References

  1. 1.

    Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn. Pergamon Press, New York (1976)

    Google Scholar 

  2. 2.

    Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, Boston (1980)

    Google Scholar 

  3. 3.

    Symon, K.R.: Mechanics, 3rd edn. Addison-Wesley, Boston (1971)

    Google Scholar 

  4. 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)

  5. 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

  6. 6.

    Gashenenko, I.N., Gorr, G.V., Kovalev, A.M.: Classical Problems of the Rigid Body Dynamics. Naukova Dumka, Kiev (2012)

    Google Scholar 

  7. 7.

    Llibre, J., Ramírez, R., Sadovskaia, N.: Integrability of the constrained rigid body. Nonlinear Dyn. 73(4), 2273–2290 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kamke, E.: Hand-book for Ordinary Differential Equations. Science, Moscow (1971)

    Google Scholar 

  9. 9.

    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)

    MathSciNet  Article  MATH  Google Scholar 

  10. 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

  11. 11.

    Popov, S.I.: On the motion of a heavy rigid body about a fixed point. Acta Mech. 85(1), 1–11 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Elmandouh, A.A.: New integrable problems in rigid body dynamics with quartic integrals. Acta Mech. 226(8), 2461–2472 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    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)

    Article  MATH  Google Scholar 

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Acknowledgements

I am thankful to Dr. Hamad H. Yehya for the insightful motivation during the fruitful discussions in the process of preparing of this manuscript.

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Correspondence to Sergey V. Ershkov.

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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

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Mathematics Subject Classification

  • 70E40 (integrable cases of motion)