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, P2 = (4π2/GM) a3 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/(2m|E|)1/2, with A = Runge-Lenz vector,

[Li, Lj] = εijk Lk ,   [Di, Lj] = εijk Dk ,   [Di, Dj] = εijk Lk .

* 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].
@ 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 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 = m1m2/(m1+m2) orbiting a fixed mass m1 + m2 at a separation r = r1 + r2.

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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