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

Euler integral and perihelion librations

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

Received  July 2019 Revised  December 2019 Published  December 2020 Early access  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.

Citation: Gabriella Pinzari. Euler integral and perihelion librations. Discrete and Continuous Dynamical Systems, 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. 

[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. 

[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.

[4]

A. BoscagginA. 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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[16]

H. WaalkensH. 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.

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. 

[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. 

[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.

[4]

A. BoscagginA. 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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[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.

[16]

H. WaalkensH. 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.

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 $
Figure 2.  Projection of the motion in the plane (g, G)
Figure 3.  Projection of the motion in the plane $ (\ell, {\Lambda})$
Figure 4.  Projection of the motion in the plane (r; R)
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