Lyapunov stability of a heavy charged gyrostat in a central attractive potential

Muneerah AL Nuwairan; Adel Elmandouh; Muneerah AL Nuwairan; Adel Elmandouh

doi:10.3934/math.2026391

AIMS Mathematics

2026, Volume 11, Issue 4: 9439-9469. doi: 10.3934/math.2026391

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Lyapunov stability of a heavy charged gyrostat in a central attractive potential

Muneerah AL Nuwairan ^,,
Adel Elmandouh

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 20 January 2026 Revised: 26 March 2026 Accepted: 31 March 2026 Published: 08 April 2026
MSC : 70E55, 37J25

This study investigated the stability of certain motions of a heavy, electrically charged gyrostat moving in a central attractive field. The Euler–Poisson equations were derived and then reformulated within the Lie–Poisson framework. We identified and analyzed the system's permanent rotations, presenting a mechanical interpretation of these motions. To assess stability, the Energy–Casimir method is employed to establish sufficient stability criteria. Moreover, the necessary conditions for the stability of the corresponding equilibrium configurations were also presented.
- Lyapunov stability,
- energy Casimir method,
- rigid body,
- gyrostat
Citation: Muneerah AL Nuwairan, Adel Elmandouh. Lyapunov stability of a heavy charged gyrostat in a central attractive potential[J]. AIMS Mathematics, 2026, 11(4): 9439-9469. doi: 10.3934/math.2026391

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Abstract

This study investigated the stability of certain motions of a heavy, electrically charged gyrostat moving in a central attractive field. The Euler–Poisson equations were derived and then reformulated within the Lie–Poisson framework. We identified and analyzed the system's permanent rotations, presenting a mechanical interpretation of these motions. To assess stability, the Energy–Casimir method is employed to establish sufficient stability criteria. Moreover, the necessary conditions for the stability of the corresponding equilibrium configurations were also presented.

References

[1]	E. Leimanis, The general problem of the motion of coupled rigid bodies about a fixed point, Springer, Berlin, 1965. https://doi.org/10.1007/978-3-642-88412-2
[2]	A. V. Borisov, I. Mamaev, Rigid body dynamics: Hamiltonian methods, integrability, chaos, Institute of Computer Science, Moscow-Izhevsk, 2005.
[3]	N. Zhukovskii, Motion of a rigid body containing a cavity filled with a homogeneous continuous liquid, Russia, Collected Works, 2 (1949), 31–152.
[4]	V. Volterra, Sur la théorie des variations des latitudes, Acta Math., 22 (1899), 201. https://doi.org/10.1007/BF02417877 doi: 10.1007/BF02417877
[5]	J. E. Cochran, P. H. Shu, S. D. Rew, Attitude motion of asymmetric dual-spin spacecraft, J. Guid. Control Dynam., 5 (1982), 37–42. https://doi.org/10.2514/3.56136 doi: 10.2514/3.56136
[6]	V. Aslanov, V. Yudintsev, Dynamics and chaos control of asymmetric gyrostat satellites, Cosmic Res., 52 (2014), 216–228. https://doi.org/10.1134/S0010952514030010 doi: 10.1134/S0010952514030010
[7]	A. Ivanov, Attenuation control of gyrostat without energy supply, International, J. NonLin. Mech., 154 (2023), 104441. https://doi.org/10.1016/j.ijnonlinmec.2023.104441 doi: 10.1016/j.ijnonlinmec.2023.104441
[8]	J. Zhao, X. Zhong, K. Yu, M. Xu, Effect of gyroscopic moments on the attitude stability of a satellite in an elliptical orbit, Nonlinear Dynam., 2023, 1–20. https://doi.org/10.21203/rs.3.rs-2475285/v1 doi: 10.21203/rs.3.rs-2475285/v1
[9]	H. M. Yehia, Rigid body dynamics: A Lagrangian approach, Springer, Nature, 45 (2022).
[10]	M. El-Borhamy, On the existence of new integrable cases for Euler-Poisson equations in Newtonian fields, Alex. Eng. J., 58 (2019), 733–744. https://doi.org/10.1016/j.aej.2019.06.004 doi: 10.1016/j.aej.2019.06.004
[11]	M. Marwan, V. D. Santos, M. Z. Abidin, A. Xiong, Coexisting attractor in a gyrostat chaotic system via basin of attraction and synchronization of two nonidentical mechanical systems, Mathematics, 10 (2022), 1914. https://doi.org/10.3390/math10111914 doi: 10.3390/math10111914
[12]	V. Tsogas, T. Kalvouridis, A. Mavraganis, Equilibrium states of a gyrostat satellite in an annular configuration of n big bodies, Acta Mech., 175 (2005), 181–195. https://doi.org/10.1007/s00707-004-0189-8 doi: 10.1007/s00707-004-0189-8
[13]	T. Kalvouridis, V. Tsogas, Rigid body dynamics in the restricted ring problem of n+1 bodies, Astrophys. Space Sci., 282 (2002), 749–763. https://doi.org/10.1023/A:1021144514396 doi: 10.1023/A:1021144514396
[14]	J. Kuang, A. Leung, S. Tan, Hamiltonian and chaotic attitude dynamics of an orbiting gyrostat satellite under gravity-gradient torques, Physica D, 186 (2003), 1–19. https://doi.org/10.1016/S0167-2789(03)00241-0 doi: 10.1016/S0167-2789(03)00241-0
[15]	A. K. Seshadri, S. Lakshmivarahan, Minimal chaotic models from the volterra gyrostat, Physica D, 456 (2023), 133948. https://doi.org/10.1016/j.physd.2023.133948 doi: 10.1016/j.physd.2023.133948
[16]	A. Farag, T. Amer, W. Amer, The periodic solutions of a symmetric charged gyrostat for a slightly relocated center of mass, Alex. Eng. J., 61 (2022), 7155–7170. https://doi.org/10.1016/j.aej.2021.12.059 doi: 10.1016/j.aej.2021.12.059
[17]	F. El-Sabaa, T. Amer, A. Sallam, I. Abady, Modeling and analysis of the nonlinear rotatory motion of an electromagnetic gyrostat, Alex. Eng. J., 61 (2022), 1625–1641. https://doi.org/10.1016/j.aej.2021.06.066 doi: 10.1016/j.aej.2021.06.066
[18]	T. Amer, W. Amer, M. A. Nuwairan, H. Elkafly, Evaluating the motion of a charged solid body having a globular cavity, Alex. Eng. J., 104 (2024), 85–94. https://doi.org/10.1016/j.aej.2024.06.031 doi: 10.1016/j.aej.2024.06.031
[19]	A. Elipe, V. Lanchares, Two equivalent problems: Gyrostats in free motion and parametric quadratic hamiltonians, Mech. Res. Commun., 24 (1997), 583–590. https://doi.org/10.1016/S0093-6413(97)00074-8 doi: 10.1016/S0093-6413(97)00074-8
[20]	J. Vera, A. Vigueras, Hamiltonian dynamics of a gyrostat in the n-body problem: Relative equilibria, Celest. Mech. Dyn. Astr., 94 (2006), 289–315. https://doi.org/10.1007/s10569-005-5910-y doi: 10.1007/s10569-005-5910-y
[21]	M. Iñarrea, V. Lanchares, A. I. Pascual, A. Elipe, Stability of the permanent rotations of an asymmetric gyrostat in a uniform newtonian field, Appl. Math. Comput., 293 (2017), 404–415. https://doi.org/10.1016/j.amc.2016.08.041 doi: 10.1016/j.amc.2016.08.041
[22]	J. Vera, The gyrostat with a fixed point in a newtonian force field: Relative equilibria and stability, J. Math. Anal. Appl., 401 (2013), 836–849. https://doi.org/10.1016/j.jmaa.2012.11.003 doi: 10.1016/j.jmaa.2012.11.003
[23]	V. Lanchares, M. Iñarrea, A. I. Pascual, A. Elipe, Stability conditions for permanent rotations of a heavy gyrostat with two constant rotors, Mathematics, 10 (2022), 1882. https://doi.org/10.3390/math10111882 doi: 10.3390/math10111882
[24]	A. Elmandouh, On the stability of the permanent rotations of a charged rigid body-gyrostat, Acta Mech., 228 (2017), 3947–3959. https://doi.org/10.1007/s00707-017-1927-z doi: 10.1007/s00707-017-1927-z
[25]	A. Elmandouh, F. H. Alsaad, The stability of certain motion of a charged gyrostat in newtonian force field, Adv. Astron., 2021 (2021), 1–11. https://doi.org/10.1155/2021/6660028 doi: 10.1155/2021/6660028
[26]	A. A. Elmandouh, A. G. Ibrahim, Hamiltonian structure, equilibria, and stability for an axisymmetric gyrostat motion in the presence of gravity and magnetic fields, Acta Mech., 230 (2019), 2539–2548. https://doi.org/10.1007/s00707-019-02413-y doi: 10.1007/s00707-019-02413-y
[27]	A. Elmandouh, On the stability of certain motions of a rigid body-gyrostat in an incompressible ideal fluid, Int. J. Nonlin. Mech., 120 (2020), 103419. https://doi.org/10.1016/j.ijnonlinmec.2020.103419 doi: 10.1016/j.ijnonlinmec.2020.103419
[28]	M. Alfadhli, A. Elmandouh, M. Al Nuwairan, Some dynamic aspects of a sextic galactic potential in a rotating reference frame, Appl. Sci., 13 (2023), 1123. https://doi.org/10.3390/app13021123 doi: 10.3390/app13021123
[29]	A. Kosov, E. Semenov, On first integrals and stability of stationary motions of gyrostat, Physica D, 430 (2022), 133103. https://doi.org/10.1016/j.physd.2021.133103 doi: 10.1016/j.physd.2021.133103
[30]	F. Crespo, E. A. Turner, Poisson structure and reduction by stages of the full gravitational n-body problem, SIAM J. Appl. Dyn. Syst., 21 (2022), 1778–1797.
[31]	J. Zapata, F. Crespo, S. Ferrer, F. Molero, Relative equilibria of an intermediary model for the roto-orbital dynamics. The low rotation regime, Adv. Space Res., 64 (2019), 1317–1330. https://doi.org/10.1016/j.asr.2019.06.033 doi: 10.1016/j.asr.2019.06.033
[32]	V. Rumyantsev, Stability of permanent rotations of a heavy rigid body, Prikl. Mat. Mekh, 20 (1956), 51–66.
[33]	V. I. Arnol'd, An a priori estimate in the theory of hydrodynamic stability, Izv. Uchebn. Zaved. Mat., 5 (1966), 3–5.
[34]	G. Pozharitskii, On the stability of permanent rotations of a rigid body with a fixed point under the action of a newtonian central force field, J. Appl. Math. Mech., 23 (1959), 1134–1137. https://doi.org/10.1016/0021-8928(59)90049-8 doi: 10.1016/0021-8928(59)90049-8
[35]	A. M. Lyapunov, The general problem of the stability of motion (in russian), Kharkov Mat. Obshch. Kharkov, 450 (1892).
[36]	E. J. Routh, Advanced dynamics of a system of rigid bodies, Dover, 1955.
[37]	L. Meirovitch, Methods of analytical dynamics, Courier Corporation, 2010.
[38]	V. V. Rumiantsev, On the stability of motion of certain types of gyrostats, J. Appl. Math. Mech., 25 (1961), 1158–1169. https://doi.org/10.1016/S0021-8928(61)80025-7 doi: 10.1016/S0021-8928(61)80025-7
[39]	M. R. C. Silva, Attitude stability of a gravity-stabilized gyrostat satellite, Celest. Mech., 2 (1970), 147–165. https://doi.org/10.1007/BF01229493 doi: 10.1007/BF01229493
[40]	T. Kane, R. Fowler, Equivalence of two gyrostatic stability problems, J. Appl. Mech., 1970. https://doi.org/10.1115/1.3408674
[41]	A. Anchev, On the stability of permanent rotations of a heavy gyrostat, J. Appl. Math. Mech., 26 (1962), 26–34. https://doi.org/10.1016/0021-8928(62)90099-0 doi: 10.1016/0021-8928(62)90099-0
[42]	A. Anchev, On the stability of permanent rotations of a quasi-symmetrical gyrostat, J. Appl. Math. Mech., 28 (1964), 194–197. https://doi.org/10.1016/0021-8928(64)90148-0 doi: 10.1016/0021-8928(64)90148-0
[43]	N. Kolesnikov, On the stability of a free gyrostat, J. Appl. Math. Mech., 27 (1963), 1063–1069. https://doi.org/10.1016/0021-8928(63)90187-4 doi: 10.1016/0021-8928(63)90187-4
[44]	D. D. Holm, J. E. Marsden, T. Ratiu, A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1–116.
[45]	M. Iñarrea, V. Lanchares, A. I. Pascual, A. Elipe, On the stability of a class of permanent rotations of a heavy asymmetric gyrostat, Regul. Chaotic Dyn., 22 (2017), 824–839. https://doi.org/10.1134/S156035471707005X doi: 10.1134/S156035471707005X
[46]	J. L. Guirao, J. A. Vera, Equilibria, stability and hamiltonian Hopf bifurcation of a gyrostat in an incompressible ideal fluid, Physica D, 241 (2012), 1648–1654. https://doi.org/10.1016/j.physd.2012.07.003 doi: 10.1016/j.physd.2012.07.003
[47]	T. C. Bradbery, Theoretical mechanics, Wiley, New York, 1968.
[48]	H. M. Yehia, Equivalent problems in rigid body dynamics-Ⅰ, Celest. Mech., 41 (1987), 275–288. https://doi.org/10.1007/BF01238764 doi: 10.1007/BF01238764
[49]	H. M. Yehia, On the motion of a rigid body acted upon by potential and gyroscopic forces, Ⅱ-A new form of the equations of motion of a rigid body in an ideal incompressible fluid, J. Theor. Appl. Mech., 5 (1986), 747–753.
[50]	P. Birtea, I. Caşu, D. Comănescu, Hamilton–poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque, Phys. Lett. A, 375 (2011), 3941–3945. https://doi.org/10.1016/j.physleta.2011.08.075 doi: 10.1016/j.physleta.2011.08.075
[51]	R. Grammel, Die stabilität der staudeschen kreiselbewegungen, Math. Z., 6 (1920), 124–142. https://doi.org/10.1007/BF01202997 doi: 10.1007/BF01202997
[52]	J. P. Ortega, T. S. Ratiu, Non-linear stability of singular relative periodic orbits in hamiltonian systems with symmetry, J. Geom. Phys., 32 (1999), 160–188. https://doi.org/10.1016/S0393-0440(99)00024-8 doi: 10.1016/S0393-0440(99)00024-8

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