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 = (4²/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,

[L_i, L_j] = _ijk L_k ,   [D_i, L_j] = _ijk D_k ,   [D_i, D_j] = _ijk L_k .

* Question: Show that the Moon's orbit around the Sun is convex [from ss].
@ General references: Vogt AJP(96) [derivation of laws]; Derbes AJP(01) [hodographic solutions]; Osler AJP(01) [first law]; Cordani 03; Kowen & Mathur AJP(03) [geometry of orbits]; Nauenberg phy/05 [history: Hooke's contributions]; Chang & Hsiang a0801 [Newton and Kepler's laws]; {& J Weinberg, SU seminar 1982}.
@ Related topics: Gergely et al ApJS(00)gq/99 [true and eccentric anomaly]; Klacka & Gajdosik ap/99 [including galactic disk]; Abramowicz & Kluzniak GRG(03)gq/02 [and general relativity]; > s.a. Bertrand's Theorem; Poynting-Robertson; Runge-Lenz; Symplectic Integrators.
@ 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 a0802 [and orbital precession].
@ Generalizations: Abramowicz et al GRG(97) [perihelion precession with curved space]; Cariñena et al mp/05, Pronko TMP(08)-a0705 [on spaces of constant curvature]; Meng mp/05.

Two-Body Problem > s.a. classical systems.
* Reduction: Can be expressed in terms of a body with the reduced mass = m₁m₂/(m₁+m₂) orbiting a fixed mass m₁ + m₂ at a separation r = r₁ + r₂.

Three-Body Problem > s.a. 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. orbits 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].
@ Periodic solutions: Chenciner & Montgomery AM(00)m.DS; Arioli CMP(02) [periodic, entropy].

Other Aspects and Generalizations
@ Celestial mechanics: Roy 05.
@ Relativistic corrections: Alaniz AJP(02) [and tests of general relativity].
@ Many-center problem: Knauf & Taimanov MA(05)m.DS/04 [integrability].
@ Gravitational assist/slingshot: Van Allen AJP(03); Dykla et al AJP(04).
@ Related topics: Meng mp/05 [MICZ-Kepler problem, any D]; Van Allen AJP(06) [asteroid encounter with planet].

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