Dynamics of Newtonian Gravitating Bodies |

**Kepler Problem**
> s.a. newtonian gravity; tests and violations.

* __Idea__: The problem of Newtonian
motion in a 1/*r* central potential, like planetary motion around the Sun.

* __Kepler's laws__:
(1) Law of ellipses, giving the shape of the orbit; (2) Law of areas, relating
the speeds at different points from conservation of angular momentum;
(3) Law of periods, *P*^{2}
= (4π^{2}/*GM*) *a*^{3}
around a fixed center of attraction of mass *M*.

* __Results__: In velocity space,
the motion follows a circle, either a complete one or just an arc.

* __Perturbed__: With the
addition of a gravitational wave it becomes a Hill system.

* __Symmetry group__:
An SO(4) algebra, generated by **L** and **D**:=
**A**/(2*m*|*E*|)^{1/2},
with **A** = Runge-Lenz vector,

[*L*_{i},
*L*_{j}]
= *ε*_{ijk}
*L*_{k} ,
[*D*_{i},
*L*_{j}]
= *ε*_{ijk}
*D*_{k} ,
[*D*_{i},
*D*_{j}]
= *ε*_{ijk}
*L*_{k} .

* __Question__: Show that
the Moon's orbit around the Sun is convex [from ss].

@ __General references__: {& J Weinberg, SU seminar 1982};
Vogt AJP(96)apr [derivation of laws];
Derbes AJP(01)apr [hodographic solutions];
Osler AJP(01)oct [first law];
Cordani 03;
Kowen & Mathur AJP(03)apr [geometry of orbits];
Nauenberg phy/05 [history: Hooke's contributions];
Chang & Hsiang a0801 [Newton and Kepler's laws];
Provost & Bracco EJP(09)-a0812 [without differential equations];
Pál MNRAS(09)-a0904 [analytical solution];
Wulfman JPA(09) [dynamical symmetries];
De Laurentis a1004;
Xu EJP(12) [simple derivation of the first law using complex variables];
Unruh a1803 [simple derivation without calculus].

@ __Perturbations__:
Chicone et al AIHP(96)gq/95,
JMP(96)gq [gravitational radiation and "ionization"];
Gergely et al in(02)gq/07;
Adkins & McDonnell PRD(07)gq,
comment Chashchina & Silagadze PRD(08)-a0802 [and orbital precession];
Lemmon & Mondragon a1012/AJP [special-relativistic corrections];
Iorio IJMPD(15) [due to the oblateness of the central body].

@ __In curved spaces__:
Abramowicz et al GRG(97),
Abramowicz a1212 [perihelion precession];
Keane et al JMP(00)-a1411 [on spaces of constant curvature];
Cariñena et al JMP(05)mp,
Pronko TMP(08)-a0705 [constant curvature];
Meng PAN(08)mp/05;
Le Tiec CQG(12)-a1202
[Kepler's third law for circular orbits, post-Newtonian generalization, and the helical Killing vector];
Witzany & Lämmerzahl ApJ(17)-a1601 [pseudo-Newtonian limit for geodesics in arbitrary spacetimes].

@ __Related topics__: Gergely et al ApJS(00)gq/99 [true and eccentric anomaly];
Klačka & Gajdošík ap/99 [including galactic disk];
Abramowicz & Kluzniak GRG(03)gq/02 [and general relativity];
Hsiang et al EJP(11)-a1105 [for the Earth];
Borghi EJP(13) [adiabatic invariants, elementary introduction];
Horvathy FS-a1404 [Kepler's laws from the harmonic oscillator].

> __Related topics__: see Bertrand's Theorem;
Kustaanheimo-Stiefel Transformation;
Poynting-Robertson Effect; Runge-Lenz Vector;
Symplectic Integrators.

**Two-Body Problem** > s.a. classical systems.

* __Reduction__: Can be expressed
in terms of a body with the reduced mass *μ*
= *m*_{1}*m*_{2}
/ (*m*_{1}+*m*_{2})
orbiting a fixed mass *m*_{1}
+ *m*_{2} at a separation
**r** = **r**_{1}
+ **r**_{2}.

**Three-Body Problem** > s.a. classical systems;
dynamics of gravitating bodies; geometric phase.

* __Idea__: A famously chaotic problem
in Newtonian gravitation.

* __Choreographic solution__: One in which
each massive particle moves periodically in a single closed orbit; One example is
a stable figure-eight orbit, first found by Moore (1993) and re-discovered with its
existence proof by Chenciner and Montgomery (2000);
> s.a. motion of gravitating objects [in general relativity].

@ __General references__:
Gutzwiller RMP(98) [Moon-Earth-Sun];
Posch & Thirring JMP(00);
Henkel PhSc(01)phy/02 [Sundman solution];
Wardell MNRAS(02)gq,
MNRAS(03)gq/02 [with radiation damping];
Mehmood et al mp/05 [closed form approximation of motion];
Šuvakov & Dmitrašinović PRL(13)-a1303
+ news sci(13)mar [13 new families of solutions].

@ __Periodic solutions__: Chenciner & Montgomery AM(00)m.DS;
Arioli CMP(02) [periodic, entropy];
Bistafa a2104 [Euler's exact syzygy solution].

@ __Other related topics__: Perdomo a1601 [relativistic restricted three-body problem, Lagrange points].

> __Online resources__:
see Scholarpedia page;
Wikipedia page.

**Other Aspects and Generalizations**
> s.a. electromagnetism [orbits of charged spheres].

@ __Celestial mechanics__:
Roy 05;
Kopeikin et al 11 [relativistic; r CP(12),
GRG(13)].

@ __Relativistic corrections__: Alaniz AJP(02)may [and tests of general relativity];
Capozziello et al PS(09)-a0812 [with gravitomagnetic corrections].

@ __Quantum-motivated corrections__: Silagadze PLA(09)-a0901 [from modified commutation relations].

@ __Two-fixed-center problem__:
Waalkens et al PhyD(04);
Biscani & Izzo MNRAS(16)-a1510 [3D, explicit, complete and closed-form solution].

@ __Many-center problem__: Knauf & Taimanov MA(05)m.DS/04 [integrability].

@ __Gravitational assist / slingshot__:
Van Allen AJP(03)may;
Dykla et al AJP(04)may.

@ __Related topics__: Meng JMP(07)mp/05 [MICZ-Kepler problem, any *D*];
Van Allen AJP(06)aug [asteroid encounter with planet].

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