A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915350896967680 |
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| author | Guardia, Marcel Paradela, Jaime Seara, Tere M. |
| author_facet | Guardia, Marcel Paradela, Jaime Seara, Tere M. |
| contents | A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3 Body Problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough.
In order to prove this result we show that a degenerate Arnold Diffusion Mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the one by Delshams, Kaloshin, De la Rosa, Seara (2019) for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called a priori stable setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06973 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem Guardia, Marcel Paradela, Jaime Seara, Tere M. Dynamical Systems A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0,m_1$. We consider the Restricted Planar Elliptic 3 Body Problem (RPE3BP) where the primaries revolve in Keplerian ellipses. We prove that the RPE3BP exhibits topological instability: for any values of the masses $m_0,m_1$ (except $m_0=m_1$), we build orbits along which the angular momentum of the massless body experiences an arbitrarily large variation provided the eccentricity of the orbit of the primaries is positive but small enough. In order to prove this result we show that a degenerate Arnold Diffusion Mechanism, which moreover involves exponentially small phenomena, takes place in the RPE3BP. Our work extends the one by Delshams, Kaloshin, De la Rosa, Seara (2019) for the a priori unstable case $m_1/m_0\ll1$, to the case of arbitrary masses $m_0,m_1>0$, where the model displays features of the so-called a priori stable setting. |
| title | A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2302.06973 |