On the analytic properties of the perturbing function in the PCR3Body Problem
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| Format: | Preprint |
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2025
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| author | Falcolini, Corrado Zaccaria, Davide |
| author_facet | Falcolini, Corrado Zaccaria, Davide |
| contents | We provide a new expansion of the Fourier coefficient of the Perturbing function of the PCR3Body problem in terms of Hansen Coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e small value of eccentricity and semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the analytic properties of the perturbing function in the PCR3Body Problem Falcolini, Corrado Zaccaria, Davide Dynamical Systems 37J05, 37J35, 37J40, 37N05, 70H05, 70H08 We provide a new expansion of the Fourier coefficient of the Perturbing function of the PCR3Body problem in terms of Hansen Coefficients. This gives us a precise asymptotic formula for the coefficient in the region of application of KAM theory (i.e small value of eccentricity and semi-major axis see e.g. \cite{Celletti-Chierchia}). Moreover, in the above region, we study the presence of zeros of the Fourier coefficient for coprime modes $(m,k) \in \Z^2$ and the presence of common zeros between coefficients relative to modes $(m,k)$,$(2m,2k)$ and $(m,k)$,$(2m,2k)$,$(3m,3k)$. Thanks to the previous expansion, this numerical analysis is done up to order $60$ in the power of eccentricity and semimajor axis. This is a first step for a possible application of \cite{Singular KAM, BBCZ} to PCR3Body Problem that would imply a reduction in terms of measure in the phase space of the so called "non--torus" set from $O(1-\sqrt{\e})$ (implied by standard KAM theory) to $O(1-\e |\log\e|^c )$ for some $c>0$. |
| title | On the analytic properties of the perturbing function in the PCR3Body Problem |
| topic | Dynamical Systems 37J05, 37J35, 37J40, 37N05, 70H05, 70H08 |
| url | https://arxiv.org/abs/2508.10621 |