Nonlinear Dynamics

, Volume 73, Issue 4, pp 2273–2290

# Integrability of the constrained rigid body

• Jaume Llibre
• Rafael Ramírez
Original Paper

## Abstract

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ)=U(γ1,γ2,γ3). This motion subject to the constraint 〈ν,ω〉=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω1,ω2,ω3) is the angular velocity and 〈 , 〉 is the inner product of $$\mathbb{R}^{3}$$.

We provide the following new integrable cases.

(i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that
$$U=-\frac{1}{2I_1I_2}\bigl(I_1\mu^2_1+I_2 \mu^2_2\bigr),$$
where I1,I2, and I3 are the principal moments of inertia of the body, μ1 and μ2 are solutions of the first-order partial differential equation
$$\gamma_3 \biggl(\frac{\partial\mu_1}{\partial\gamma_2}- \frac{\partial\mu_2}{\partial \gamma_1} \biggr)- \gamma_2\frac{\partial \mu_1}{\partial\gamma_3}+\gamma_1\frac{\partial\mu_2}{\partial \gamma_3}=0.$$
(ii) The Veselova problem is integrable for the potential
$$U=-\frac{\varPsi^2_1+\varPsi^2_2}{2(I_1\gamma^2_2+I_2\gamma^2_1)},$$
where Ψ1 and Ψ2 are the solutions of the first-order partial differential equation where $$p=\sqrt{I_{1}I_{2}I_{3} (\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}} )}$$.
Also it is integrable when the potential U is a solution of the second-order partial differential equation where $$\tau_{2}=I_{1}\gamma^{2}_{1}+I_{2}\gamma^{2}_{2}+I_{3}\gamma^{2}_{3}$$ and $$\tau_{3}=\frac{\gamma^{2}_{1}}{I_{1}}+\frac{\gamma^{2}_{2}}{I_{2}}+ \frac{\gamma^{2}_{3}}{I_{3}}$$.

Moreover, we show that these integrable cases contain as a particular case the previous known results.

### Keywords

Ordinary differential equation Invariant measure Mechanical systems Nonholonomic system Constraint Rigid body Suslov problem Veselova problem Integrability

## Notes

### Acknowledgements

J. Llibre is partially supported by a MINECO/FEDER grant number MTM2009-03437, an AGAUR grant number 2009SGR-410, ICREA Academia and two FPZ-PEOPLE-2012-IRSES numbers 316338 and 318999. R. Ramírez was partly supported by the Spanish Ministry of Education through projects TSI2007-65406-C03-01 “E-AEGIS” and Consolider CSD2007-00004 “ARES”.

### References

1. 1.
Arnold, V.I.: Dynamical Systems III. Springer, Berlin (1996) Google Scholar
2. 2.
Borisov, A.V., Mamaev, I.S.: Dynamics of a Rigid Body. Institute of Computer Science, Moscow (2005) (in Russian)
3. 3.
Borisov, A.V., Mamaev, I.S.: Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems. Regul. Chaotic Dyn. 13, 443–490 (2008)
4. 4.
Dragović, V., Gajić, B., Jovanović, B.: Generalizations of classical integrable nonholonomic rigid body systems. J. Phys. A, Math. Gen. 31, 9861–9869 (1998)
5. 5.
Fedorov, Y.N., Maciejewski, A., Przybylska, M.: The Poisson equations in the nonholonomic Suslov problem: integrability, meromorphic and hypergeometric solutions. J. Nonlinear Sci. 22, 2231–2259 (2009)
6. 6.
Fedorov, Y.N., Jovanovic, B.: Nonholonomic LR systems as generalized Chaplygin systems with an invariant measure and flows on homogeneous space. J. Nonlinear Sci. 14, 341–381 (2004)
7. 7.
Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advances Series in Nonlinear Dynamics, vol. 19. World Scientific, Singapore (2001)
8. 8.
Kharlamova-Zabelina, E.I.: Rapid rotation of a rigid body about a fixed point under the presence of a nonholonomic constraints. Vestn. Mosk. Univ., Ser. Mat. Mekh. Astron. Fiz. Khim. 6, 25–34 (1957) (Russian) Google Scholar
9. 9.
Kozlov, V.V.: On the integration theory of the equations of nonholonomic mechanics. Regul. Chaotic Dyn. 7, 161–172 (2002)
10. 10.
Kozlov, V.V.: Symmetries, Topology and Resonances in Hamiltonian Mechanics. Springer, Berlin (1995) Google Scholar
11. 11.
Maciejewski, A.J., Przybylska, M.: Non-integrability of the Suslov problem. Regul. Chaotic Dyn. 7, 73–80 (2002)
12. 12.
Okuneva, G.G.: A qualitative investigation of an integrable variant of the nonholonomic Suslov problem. Vestn. Mosk. Univ., Ser. Mat. Mekh. 5, 59–64 (1987) (Russian)
13. 13.
Ramirez, R., Sadovskaia, N.: Cartesian approach for constrained mechanical systems. Rep. Math. Phys. 65(1), 109–139 (2010)
14. 14.
Suslov, G.K.: Theoretical Mechanics. Gostehizdat, Moscow (1946) Google Scholar
15. 15.
Veselova, L.E.: Novie sluchai integriruemosti uravneniy dvijenia tverdogo tela pri nalichii negolonomnoi sviazi. Sbornik Geometria, dif. uravnenia i mekhanika, MGU, pp. 64–68 (1986) (in Russian) Google Scholar

## Authors and Affiliations

• Jaume Llibre
• 1
• Rafael Ramírez
• 2