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December  2020, 40(12): 6919-6943. doi: 10.3934/dcds.2020165

Euler integral and perihelion librations

Gabriella Pinzari

Dipartimento di Matematica "Tullio Levi–Civita", via Trieste, 63, 35121 Padova, Italy

Received  July 2019 Revised  December 2019 Published  March 2020

Fund Project: The author is supported by the European Research Council. Grant 677793 Stable and Chaotic Motions in the Planetary Problem

We discuss dynamical aspects of an analysis of the two–centre problem started in [15]. The perturbative nature of our approach allows us to foresee applications to the three–body problem.

Keywords: Two–centre problem, Euler integral, canonical coordinates, three–body problem.
Mathematics Subject Classification: Primary: 34C20, 70F07, 37J10, 37J15, 37J35; Secondary: 34D10, 70F10, 70F15, 37J25, 37J40.
Citation: Gabriella Pinzari. Euler integral and perihelion librations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6919-6943. doi: 10.3934/dcds.2020165
References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192.   Google Scholar

[2]

A. A. Bekov and T. B. Omarov, Integrable cases of the Hamilton-Jacobi equation and some nonsteady problems of celestial mechanics, Soviet Astronomy, 22 (1978), 366-370.   Google Scholar

[3]

F. Biscani and D. Izzo, A complete and explicit solution to the three-dimensional problem of two fixed centres, Monthly Notices Roy. Astronomical Soc., 455 (2016), 3480-3493.  doi: 10.1093/mnras/stv2512.  Google Scholar

[4]

A. Boscaggin, A. Dambrosio and S. Terracini, Scattering parabolic solutions for the spatial $N$-centre problem, Arch. Rational Mech. Anal., 223 (2017), 1269-1306.  doi: 10.1007/s00205-016-1057-0.  Google Scholar

[5]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised, Celestial Mech. Dynam. Astronom., 109 (2011), 285-301.  doi: 10.1007/s10569-010-9329-8.  Google Scholar

[6]

L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori, Invent. Math., 186 (2011), 1-77.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[7]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195.  doi: 10.1007/BF01234305.  Google Scholar

[8]

H. R. Dullin and R. Montgomery, Syzygies in the two center problem, Nonlinearity, 29 (2016), 1212-1237.  doi: 10.1088/0951-7715/29/4/1212.  Google Scholar

[9]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582.  doi: 10.1017/S0143385704000410.  Google Scholar

[10]

C. G. J. Jacobi, Sur l'élimination des noeuds dans le problème des trois corps, J. Reine Angew. Math., 26 (1843), 115-131.  doi: 10.1515/crll.1843.26.115.  Google Scholar

[11]

C. G. J. Jacobi, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, 51, Hindustan Book Agency, New Delhi, 2009.  Google Scholar

[12]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom., 62 (1995), 193-217.  doi: 10.1007/BF00692088.  Google Scholar

[13]

G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906.  doi: 10.1134/S1560354713060178.  Google Scholar

[14]

G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem, Mem. Amer. Math. Soc., 255 (2018). doi: 10.1090/memo/1218.  Google Scholar

[15]

G. Pinzari, A first integral to the partially averaged Newtonian potential of the three-body problem, Celestial Mech. Dynam. Astronom., 131 (2019), 30pp. doi: 10.1007/s10569-019-9899-z.  Google Scholar

[16]

H. Waalkens, H. R. Dullin and P. H. Richter, The problem of two fixed centers: Bifurcations, actions, monodromy, Phys. D, 196 (2004), 265-310.  doi: 10.1016/j.physd.2004.05.006.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192.   Google Scholar

[2]

A. A. Bekov and T. B. Omarov, Integrable cases of the Hamilton-Jacobi equation and some nonsteady problems of celestial mechanics, Soviet Astronomy, 22 (1978), 366-370.   Google Scholar

[3]

F. Biscani and D. Izzo, A complete and explicit solution to the three-dimensional problem of two fixed centres, Monthly Notices Roy. Astronomical Soc., 455 (2016), 3480-3493.  doi: 10.1093/mnras/stv2512.  Google Scholar

[4]

A. Boscaggin, A. Dambrosio and S. Terracini, Scattering parabolic solutions for the spatial $N$-centre problem, Arch. Rational Mech. Anal., 223 (2017), 1269-1306.  doi: 10.1007/s00205-016-1057-0.  Google Scholar

[5]

L. Chierchia and G. Pinzari, Deprit's reduction of the nodes revised, Celestial Mech. Dynam. Astronom., 109 (2011), 285-301.  doi: 10.1007/s10569-010-9329-8.  Google Scholar

[6]

L. Chierchia and G. Pinzari, The planetary $N$-body problem: Symplectic foliation, reductions and invariant tori, Invent. Math., 186 (2011), 1-77.  doi: 10.1007/s00222-011-0313-z.  Google Scholar

[7]

A. Deprit, Elimination of the nodes in problems of $n$ bodies, Celestial Mech., 30 (1983), 181-195.  doi: 10.1007/BF01234305.  Google Scholar

[8]

H. R. Dullin and R. Montgomery, Syzygies in the two center problem, Nonlinearity, 29 (2016), 1212-1237.  doi: 10.1088/0951-7715/29/4/1212.  Google Scholar

[9]

J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergodic Theory Dynam. Systems, 24 (2004), 1521-1582.  doi: 10.1017/S0143385704000410.  Google Scholar

[10]

C. G. J. Jacobi, Sur l'élimination des noeuds dans le problème des trois corps, J. Reine Angew. Math., 26 (1843), 115-131.  doi: 10.1515/crll.1843.26.115.  Google Scholar

[11]

C. G. J. Jacobi, Jacobi's Lectures on Dynamics, Texts and Readings in Mathematics, 51, Hindustan Book Agency, New Delhi, 2009.  Google Scholar

[12]

J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom., 62 (1995), 193-217.  doi: 10.1007/BF00692088.  Google Scholar

[13]

G. Pinzari, Aspects of the planetary Birkhoff normal form, Regul. Chaotic Dyn., 18 (2013), 860-906.  doi: 10.1134/S1560354713060178.  Google Scholar

[14]

G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem, Mem. Amer. Math. Soc., 255 (2018). doi: 10.1090/memo/1218.  Google Scholar

[15]

G. Pinzari, A first integral to the partially averaged Newtonian potential of the three-body problem, Celestial Mech. Dynam. Astronom., 131 (2019), 30pp. doi: 10.1007/s10569-019-9899-z.  Google Scholar

[16]

H. Waalkens, H. R. Dullin and P. H. Richter, The problem of two fixed centers: Bifurcations, actions, monodromy, Phys. D, 196 (2004), 265-310.  doi: 10.1016/j.physd.2004.05.006.  Google Scholar

Figure 1.  The phase portrait of $ {{\rm{E}}}_0 $ in the plane $ ({\rm g}, {{\rm{G}}}) $. Left: $ 0< {\delta}<1 $; Center: $ 0< {\delta}<1 $; Right: $ {\delta}>2 $
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Figure 2.  Projection of the motion in the plane (g, G)
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Figure 3.  Projection of the motion in the plane $ (\ell, {\Lambda})$
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Figure 4.  Projection of the motion in the plane (r; R)
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