Bifurcation from periodic solutions of central force problems in the three-dimensional space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909641013723136 |
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| author | Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio |
| author_facet | Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio |
| contents | The paper deals with electromagnetic perturbations of a central force problem of the form \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t} \bigl( φ(\dot{x}) \bigr) = V'(|x|) \dfrac{x}{|x|} + E_{\varepsilon}(t,x)+\dot{x} \wedge B_{\varepsilon}(t,x), \qquad x \in \mathbb{R}^3 \setminus \{0\}, \end{equation*} where $V \colon (0,+\infty) \to \mathbb{R}$ is a smooth function, $E_\varepsilon$ and $B_\varepsilon$ are respectively the electric field and the magnetic field, smooth and periodic in time, $\varepsilon\in\mathbb{R}$ is a small parameter. The considered differential operator includes, as special cases, the classical one, $φ(v)=mv$, as well as that of special relativity, $φ(v) = mv/\sqrt{1-\vert v \vert^2/c^2}$. We investigate whether non-circular periodic solutions of the unperturbed problem (i.e., with $\varepsilon=0$) can be continued into periodic solutions for $\varepsilon\neq0$ small, both for the fixed-period problem and, if the perturbation is time-independent, for the fixed-energy problem. The proof is based on an abstract bifurcation theorem of variational nature, which is applied to suitable Hamiltonian action functionals. In checking the required non-degeneracy conditions we take advantage of the existence of partial action-angle coordinates as provided by the Mishchenko--Fomenko theorem for superintegrable systems. Physically relevant problems to which our results can be applied are homogeneous central force problems in classical mechanics and the Kepler problem in special relativity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05842 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcation from periodic solutions of central force problems in the three-dimensional space Boscaggin, Alberto Feltrin, Guglielmo Papini, Duccio Dynamical Systems Mathematical Physics 34C25, 37J20, 70H12, 70H40 The paper deals with electromagnetic perturbations of a central force problem of the form \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t} \bigl( φ(\dot{x}) \bigr) = V'(|x|) \dfrac{x}{|x|} + E_{\varepsilon}(t,x)+\dot{x} \wedge B_{\varepsilon}(t,x), \qquad x \in \mathbb{R}^3 \setminus \{0\}, \end{equation*} where $V \colon (0,+\infty) \to \mathbb{R}$ is a smooth function, $E_\varepsilon$ and $B_\varepsilon$ are respectively the electric field and the magnetic field, smooth and periodic in time, $\varepsilon\in\mathbb{R}$ is a small parameter. The considered differential operator includes, as special cases, the classical one, $φ(v)=mv$, as well as that of special relativity, $φ(v) = mv/\sqrt{1-\vert v \vert^2/c^2}$. We investigate whether non-circular periodic solutions of the unperturbed problem (i.e., with $\varepsilon=0$) can be continued into periodic solutions for $\varepsilon\neq0$ small, both for the fixed-period problem and, if the perturbation is time-independent, for the fixed-energy problem. The proof is based on an abstract bifurcation theorem of variational nature, which is applied to suitable Hamiltonian action functionals. In checking the required non-degeneracy conditions we take advantage of the existence of partial action-angle coordinates as provided by the Mishchenko--Fomenko theorem for superintegrable systems. Physically relevant problems to which our results can be applied are homogeneous central force problems in classical mechanics and the Kepler problem in special relativity. |
| title | Bifurcation from periodic solutions of central force problems in the three-dimensional space |
| topic | Dynamical Systems Mathematical Physics 34C25, 37J20, 70H12, 70H40 |
| url | https://arxiv.org/abs/2506.05842 |