Dynamics and integrability of the swinging Atwood machine generalisations

Szumiński, Wojciech; Maciejewski, Andrzej J.

doi:10.1007/s11071-022-07680-4

Cited by 5 publications

(4 citation statements)

References 44 publications

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“…The literature on the existence and nonexistence of first integrals or generally the integrability problem of a dynamical system is one of the main open problems in the qualitative theory of differential systems, see [1,2,3,4]. Many non-linear dynamical systems arise in physical and electrical engineering, etc.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Integrability and dynamics Analysis of the Chaos Laser System

Amen

2023

Scientia Iranica

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In this article the complex dynamics of a laser model, which externally injected class 𝐵 which is described by a system of three nonlinear ordinary differential equations with two parameters for field intensity phase and population inversion, are studied. In particular, we investigate the integrability and nonintegrabilty of laser system in three dimension. We prove that system is completely integrable only when the parameters are zero. Particularly, we study polynomial, rational, Darboux and analytic first integrals of the mentioned system. Moreover, we compute all the invariant algebraic surfaces and exponential factors of this system. We find sufficient conditions for the existence of periodic orbits emanating from an equilibrium point origin of a laser differential system with a first integral.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Integrability and dynamics Analysis of the Chaos Laser System

Amen

2023

Scientia Iranica

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“…Because of this, many new integrable and super-integrable systems were found [64][65][66]. To exemplify it, we mention two generalizations of the swinging Atwood machine model recently studied in [24,28]. In this paper, the authors performed a detailed integrability analysis and found integrable and super-integrable cases with additional first integrals quadratic and quartic in the momenta.…”

Section: Theorem 11 (Morales-ramis (1999)) If a Hamiltonian System Is...mentioning

confidence: 99%

“…Currently, there is great activity studying variable-length pendulum systems, such as the swinging Atwood machine [27] and its generalizatins [28], the variable-length coupled pendulums recently studied in [24], or the double variable-length pendulum with counterweight mass [71][72][73]. Variable-length pendulum systems are excellent examples for studying nonlinear dynamics, chaos, and bifurcations.…”

Section: Theorem 11 (Morales-ramis (1999)) If a Hamiltonian System Is...mentioning

confidence: 99%

“…Indeed, one can find numerous papers, books, and video clips concerning their non-linear and chaotic dynamics. To exemplify it, let us mention the paradigm models such as the spring pendulum [2][3][4][5][6], the magnetic pendulum [7][8][9], the double and the triple pendulums [10][11][12][13][14][15][16][17][18][19], the coupled pendulums [20][21][22][23][24] and the swinging Atwood machine [25][26][27][28][29]. These models have been extensively studied by researchers both theoretically and practically [29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

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Dynamics and non-integrability of the double spring pendulum

Szumiński,

Maciejewski

2024

Journal of Sound and Vibration

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Chaos and integrability of relativistic homogeneous potentials in curved space

Szumiński,

Przybylska,

Maciejewski

2024

Nonlinear Dyn

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Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential $$V(\varvec{q})$$ V ( q ) . In the limit of weak potential $$2V(\varvec{q})/mc^2\ll 1$$ 2 V ( q ) / m c 2 ≪ 1 , and small momentum $$|\varvec{p} |/ mc\ll 1$$ | p | / m c ≪ 1 , these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential $$V(\varvec{q})$$ V ( q ) . We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector $$\varvec{d}$$ d satisfying $$V'(\varvec{d})=\gamma \varvec{d}$$ V ′ ( d ) = γ d for a certain $$\gamma \ne 0$$ γ ≠ 0 . They are very strong: if the relativistic system is integrable in the Liouville sense, then either $$k=\pm 2$$ k = ± 2 , or all non-trivial eigenvalues of the re-scaled Hessian $$\gamma ^{-1}V''(\varvec{d})$$ γ - 1 V ′ ′ ( d ) are either 0, or 1. Certain integrable relativistic systems are presented.

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Dynamics and integrability of the swinging Atwood machine generalisations

Cited by 5 publications

References 44 publications

Integrability and dynamics Analysis of the Chaos Laser System

Integrability and dynamics Analysis of the Chaos Laser System

Dynamics and non-integrability of the double spring pendulum

Chaos and integrability of relativistic homogeneous potentials in curved space

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