Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems
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| Format: | Preprint |
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2024
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| _version_ | 1866913356428869632 |
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| author | Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio |
| author_facet | Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio |
| contents | The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -α\frac{x}{|x|^{3}} + \varepsilon \, \nabla_{x} U(t,x), \qquad x \in \mathbb{R}^d\setminus\{0\}, \end{equation*} with $d=2$ or $d=3$, bifurcating, for $\varepsilon$ small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that $U$ is $T$-periodic in time) and the case of the fixed-energy problem (assuming that $U$ is independent of time) are considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio Dynamical Systems Mathematical Physics 34C25, 70H12, 70H40 The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -α\frac{x}{|x|^{3}} + \varepsilon \, \nabla_{x} U(t,x), \qquad x \in \mathbb{R}^d\setminus\{0\}, \end{equation*} with $d=2$ or $d=3$, bifurcating, for $\varepsilon$ small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that $U$ is $T$-periodic in time) and the case of the fixed-energy problem (assuming that $U$ is independent of time) are considered. |
| title | Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems |
| topic | Dynamical Systems Mathematical Physics 34C25, 70H12, 70H40 |
| url | https://arxiv.org/abs/2405.11189 |