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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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 |
We discuss dynamical aspects of an analysis of the two–centre problem started in [
Mathematics Subject Classification:
Primary: 34C20, 70F07, 37J10, 37J15, 37J35; Secondary: 34D10, 70F10, 70F15, 37J25, 37J40.
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. 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. |
| [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. 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. |
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. 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. |
| [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. 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. |
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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