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Blesa F, Bernal JD, Seoane JM, Sanjuán MAFM. Relativistic chaotic scattering: Unveiling scaling laws for trapped trajectories. Phys Rev E 2024; 109:044204. [PMID: 38755803 DOI: 10.1103/physreve.109.044204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2023] [Accepted: 03/19/2024] [Indexed: 05/18/2024]
Abstract
In this paper we study different types of phase space structures which appear in the context of relativistic chaotic scattering. By using the relativistic version of the Hénon-Heiles Hamiltonian, we numerically study the topology of different kind of exit basins and compare it with the case of low velocities in which the Newtonian version of the system is valid. Specifically, we numerically study the escapes in the phase space, in the energy plane, and in the β plane, which richly characterize the dynamics of the system. In all cases, fractal structures are present, and the escaping dynamics is characterized. In every case a scaling law is numerically obtained in which the percentage of the trapped trajectories as a function of the relativistic parameter β and the energy is obtained. Our work could be useful in the context of charged particles which eventually can be trapped in the magnetosphere, where the analysis of these structures can be relevant.
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Affiliation(s)
- Fernando Blesa
- Departamento de Física Aplicada, University of Zaragoza, 50009 Zaragoza, Spain
| | - Juan D Bernal
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Jesús M Seoane
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Miguel A F M Sanjuán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
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Przybylska M, Szumiński W, Maciejewski AJ. Destructive relativity. CHAOS (WOODBURY, N.Y.) 2023; 33:063156. [PMID: 37368038 DOI: 10.1063/5.0140633] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2022] [Accepted: 05/22/2023] [Indexed: 06/28/2023]
Abstract
The description of dynamics for high-energy particles requires an application of the special relativity theory framework, and analysis of properties of the corresponding equations of motion is very important. Here, we analyze Hamilton equations of motion in the limit of weak external field when potential satisfies the condition 2V(q)≪mc2. We formulate very strong necessary integrability conditions for the case when the potential is a homogeneous function of coordinates of integer non-zero degrees. If Hamilton equations are integrable in the Liouville sense, then eigenvalues of the scaled Hessian matrix γ-1V″(d) at any non-zero solution d of the algebraic system V'(d)=γd must be integer numbers of appropriate form depending on k. As it turns out, these conditions are much stronger than those for the corresponding non-relativistic Hamilton equations. According to our best knowledge, the obtained results are the first general integrability necessary conditions for relativistic systems. Moreover, a relation between the integrability of these systems and corresponding non-relativistic systems is discussed. The obtained integrability conditions are very easy to use because the calculations reduce to linear algebra. We show their strength in the example of Hamiltonian systems with two degrees of freedom with polynomial homogeneous potentials. It seems that the only integrable relativistic systems with such potentials are those depending only on one coordinate or having a radial form.
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Affiliation(s)
- Maria Przybylska
- Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland
| | - Wojciech Szumiński
- Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland
| | - Andrzej J Maciejewski
- Janusz Gil Institute of Astronomy, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra, Poland
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Bernal JD, Seoane JM, Vallejo JC, Huang L, Sanjuán MAF. Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem. Phys Rev E 2020; 102:042204. [PMID: 33212716 DOI: 10.1103/physreve.102.042204] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2020] [Accepted: 09/22/2020] [Indexed: 11/07/2022]
Abstract
The Sitnikov problem is a classical problem broadly studied in physics which can represent an illustrative example of chaotic scattering. The relativistic version of this problem can be attacked by using the post-Newtonian formalism. Previous work focused on the role of the gravitational radius λ on the phase space portrait. Here we add two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena. First, we uncover a metamorphosis of the KAM islands for which the escape regions change insofar as λ increases. Second, there are two inflection points in the unpredictability of the final state of the system when λ≃0.02 and λ≃0.028. We analyze these inflection points in a quantitative manner by using the basin entropy. This work can be useful for a better understanding of the Sitnikov problem in the context of relativistic chaotic scattering. In addition, the described techniques can be applied to similar real systems, such as binary stellar systems, among others.
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Affiliation(s)
- Juan D Bernal
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Jesús M Seoane
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Juan C Vallejo
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.,Joint Center for Ultraviolet Astronomy, AEGORA Research Group, Universidad Complutense de Madrid, Avda Puerta de Hierro s/n, 28040 Madrid, Spain
| | - Liang Huang
- Institute of Computational Physics and Complex Systems, School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
| | - Miguel A F Sanjuán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.,Department of Applied Informatics, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania
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Fernández DS, López ÁG, Seoane JM, Sanjuán MAF. Transient chaos under coordinate transformations in relativistic systems. Phys Rev E 2020; 101:062212. [PMID: 32688505 DOI: 10.1103/physreve.101.062212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2020] [Accepted: 06/15/2020] [Indexed: 06/11/2023]
Abstract
We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering to study the Lorentz factor effects on its transient chaotic dynamics. In particular, we focus on how time dilation occurs within the scattering region by measuring the time with a clock attached to the particle. We observe that the several events of time dilation that the particle undergoes exhibit sensitivity to the initial conditions. However, the structure of the singularities appearing in the escape time function remains invariant under coordinate transformations. This occurs because the singularities are closely related to the chaotic saddle. We then demonstrate using a Cantor-like set approach that the fractal dimension of the escape time function is relativistic invariant. In order to verify this result, we compute by means of the uncertainty dimension algorithm the fractal dimensions of the escape time functions as measured with an inertial frame and a frame comoving with the particle. We conclude that, from a mathematical point of view, chaotic transient phenomena are equally predictable in any reference frame and that transient chaos is coordinate invariant.
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Affiliation(s)
- D S Fernández
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - Á G López
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - J M Seoane
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | - M A F Sanjuán
- Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
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