Differential Galois approach to the non-integrability of the heavy top problem

Maciejewski, Andrzej J.; Przybylska, Maria

doi:10.5802/afst.1090

Cited by 16 publications

(12 citation statements)

References 30 publications

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“…If the Hamiltonian system (1.1) is real-meromorphically integrable in U R , then its complexification is also meromorphically integrable in U C . Such real-meromorphically nonintegrable Hamiltonian systems were also discussed by using a different approach in [13,14,32]. (iii) If x + = x − , then condition (2.7) automatically holds, so that conclusion (2.8) is necessary for the real-meromorphic integrability of (1.1).…”

Section: )mentioning

confidence: 99%

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Yagasaki

Yamanaka

2019

SIGMA

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We consider a class of two-degree-of-freedom Hamiltonian systems with saddlecenters connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.

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Section: )mentioning

confidence: 99%

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Yagasaki

Yamanaka

2019

SIGMA

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“…Remark: When comparing our system (1.2) with its "twin brother"-the EulerPoisson equations of heavy rigid body motion (see [2,3,14,15,18]) we conclude from [21] (see also [11]) that for these equations the exact counterpart of Theorem 1.1 holds. Nevertheless, the exact counterpart of Theorem 3.1 for EulerPoisson equations fails.…”

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confidence: 85%

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

Popov

Strelcyn

2008

Isr. J. Math.

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We consider the Euler equations on the Lie algebra so(4, C) with a diagonal quadratic Hamiltonian. It is known that this system always admits three functionally independent polynomial first integrals. We prove that if the system has a rational first integral functionally independent of the known three ones so called fourth integral, then it has a polynomial first integral that is also functionally independent of them. This is a consequence of more general fact that for these systems the existence of Darboux polynomial with no vanishing cofactor implies the existence of polynomial fourth integral.

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“…Recall that there are only three cases when there exists an additional first integral (and the vector field X H is completely integrable in the Liouville-Arnold sense, see [4], [27], and [20]):…”

Section: Rados Law Kurek Pawe L Lubowiecki and Henrykżo La Dekmentioning

confidence: 99%

The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

Kurek¹,

Lubowiecki²,

Żołądek³

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.

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Differential Galois approach to the non-integrability of the heavy top problem

Cited by 16 publications

References 30 publications

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

On rational integrability of Euler equations on Lie algebra so(4, ℂ)

The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

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