Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation
The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anom...
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| Format: | Article |
| Language: | English |
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Elsevier
2017-05-01
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| Series: | Theoretical and Applied Mechanics Letters |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2095034917300776 |
| _version_ | 1811339738985005056 |
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| author | Sergey V. Ershkov |
| author_facet | Sergey V. Ershkov |
| author_sort | Sergey V. Ershkov |
| collection | DOAJ |
| description | The main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ordinary differential equation (ODE) of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold, 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e→0, p=1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii’s equation. |
| first_indexed | 2024-04-13T18:31:03Z |
| format | Article |
| id | doaj.art-04021b2496bf4096af0bee02bcf54f1b |
| institution | Directory Open Access Journal |
| issn | 2095-0349 |
| language | English |
| last_indexed | 2024-04-13T18:31:03Z |
| publishDate | 2017-05-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Theoretical and Applied Mechanics Letters |
| spelling | doaj.art-04021b2496bf4096af0bee02bcf54f1b2022-12-22T02:35:04ZengElsevierTheoretical and Applied Mechanics Letters2095-03492017-05-017317517810.1016/j.taml.2017.05.005Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotationSergey V. ErshkovThe main objective for this research was the analytical exploration of the dynamics of planar satellite rotation during the motion of an elliptical orbit around a planet. First, we revisit the results of J. Wisdom et al. (1984), in which, by the elegant change of variables (considering the true anomaly f as the independent variable), the governing equation of satellite rotation takes the form of an Abel ordinary differential equation (ODE) of the second kind, a sort of generalization of the Riccati ODE. We note that due to the special character of solutions of a Riccati-type ODE, there exists the possibility of sudden jumping in the magnitude of the solution at some moment of time. In the physical sense, this jumping of the Riccati-type solutions of the governing ODE could be associated with the effect of sudden acceleration/deceleration in the satellite rotation around the chosen principle axis at a definite moment of parametric time. This means that there exists not only a chaotic satellite rotation regime (as per the results of J. Wisdom et al. (1984)), but a kind of gradient catastrophe (Arnold, 1992) could occur during the satellite rotation process. We especially note that if a gradient catastrophe could occur, this does not mean that it must occur: such a possibility depends on the initial conditions. In addition, we obtained asymptotical solutions that manifest a quasi-periodic character even with the strong simplifying assumptions e→0, p=1, which reduce the governing equation of J. Wisdom et al. (1984) to a kind of Beletskii’s equation.http://www.sciencedirect.com/science/article/pii/S2095034917300776Beletskii’s equationSatellite rotationAbel ODEGradient catastrophe |
| spellingShingle | Sergey V. Ershkov Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation Theoretical and Applied Mechanics Letters Beletskii’s equation Satellite rotation Abel ODE Gradient catastrophe |
| title | Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation |
| title_full | Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation |
| title_fullStr | Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation |
| title_full_unstemmed | Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation |
| title_short | Revolving scheme for solving a cascade of Abel equations in dynamics of planar satellite rotation |
| title_sort | revolving scheme for solving a cascade of abel equations in dynamics of planar satellite rotation |
| topic | Beletskii’s equation Satellite rotation Abel ODE Gradient catastrophe |
| url | http://www.sciencedirect.com/science/article/pii/S2095034917300776 |
| work_keys_str_mv | AT sergeyvershkov revolvingschemeforsolvingacascadeofabelequationsindynamicsofplanarsatelliterotation |