Dynamics
of Newtonian Gravitating Bodies| 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 = (4
2/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]
= 
ijk Lk , [Di,
Lj] = ijk
Dk , [Di,
Dj] = ijk
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-a0904 [analytical
solution]; Wulfman JPA(09) [dynamical symmetries].
@ 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 PRD(08)-a0802 [and
orbital precession].
@ In curved spaces:
Abramowicz et al GRG(97)
[perihelion precession]; Cariñena et al mp/05,
Pronko TMP(08)-a0705 [constant
curvature]; Meng PAN(08)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 > s.a. electromagnetism [orbits of charged spheres]
@ Celestial mechanics: Roy 05.
@ 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].
@ 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].
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16
jul
2009