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 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].

> __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].

@ __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].

@ __Many-center problem__: Knauf & Taimanov MA(05)m.DS/04 [integrability];
Biscani & Izzo MNRAS(16)-a1510 [3D two-fixed-center problem: explicit, complete and closed-form solution].

@ __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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send feedback and suggestions to bombelli at olemiss.edu – modified 19 mar 2018