Anisotropic Kepler and anisotropic two fixed centres problems

Self Cite

In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

Section: Introductionmentioning

confidence: 99%

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

Szumiński

Maciejewski

2021

Self Cite

Dynamics and integrability of the swinging Atwood machine generalisations

Szumiński

Maciejewski

2022

This paper studies the dynamics and integrability of two generalisations of a 3D Swinging Atwood’s Machine with additional Coulomb’s interactions and Hooke’s law of elasticity. The complexity of these systems is presented with the help of Poincaré cross sections, phase-parametric diagrams and Lyapunov exponents spectrums. Amazingly, such systems possess both chaotic and integrable dynamics. For the integrable cases we find additional first integrals and we construct general solutions written in terms of elliptic functions. Moreover, we present bifurcation diagrams for the integrable cases and we find resonance curves, which give families of periodic orbits of the systems. In the absence of the gravity, both models are super-integrable.

Chaos and integrability of relativistic homogeneous potentials in curved space

Szumiński,

Przybylska,

Maciejewski

2024

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.