scholarly journals Canonical Variables of The Second Kind and The Reduction of The N-Body Problem

1999 ◽  
Vol 172 ◽  
pp. 269-280
Author(s):  
John G. Bryant
Keyword(s):  
Hamiltonian System ◽  
Body Problem ◽  
Blow Up ◽  
Kepler Problem ◽  
Geometrical Optics ◽  
Canonical Variables ◽  
N Body Problem ◽  
Homogeneous Reduction ◽  
New Variables

AbstractWe introduce a new kind of canonical variables that prove very useful when the order of a Hamiltonian system can be reduced by one, as in the case of isoenergetic reduction, and of what we call homogeneous reduction. The Kepler Problem, Geometrical Optics and McGehee Blow-up are discussed as examples. Finally we carry out the isoenergetic reduction of the general N-Body Problem using the new variables, and briefly discuss its application to the problem of collision.

scholarly journals On frequency estimations in phase fitted variational integrators for the general N-body problem

PAMM ◽  
2013 ◽  
Vol 13 (1) ◽  
pp. 33-34
Author(s):  
Odysseas Kosmas ◽  
Sigrid Leyendecker
Keyword(s):  
Body Problem ◽  
Variational Integrators ◽  
N Body Problem

GLOBAL BEHAVIOR OF THE CHARGED ISOSCELES THREE-BODY PROBLEM

1994 ◽  
Vol 04 (04) ◽  
pp. 865-884 ◽  
Author(s):  
PAU ATELA ◽  
ROBERT I. McLACHLAN
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Body Problem ◽  
Kepler Problem ◽  
Global Bifurcation ◽  
Three Body Problem ◽  
Anisotropic Kepler Problem ◽  
The One ◽  
Three Body ◽  
Limiting Value

We study the global bifurcation diagram of the two-parameter family of ODE’s that govern the charged isosceles three-body problem. (The classic isosceles three-body problem and the anisotropic Kepler problem (two bodies) are included in the same family.) There are two major sources of periodic orbits. On the one hand the “Kepler” orbit, a stable orbit exhibiting the generic bifurcations as the multiplier crosses rational values. This orbit turns out to be the continuation of the classical circular Kepler orbit. On the other extreme we have the collision-ejection orbit which exhibits an “infinite-furcation.” Up to a limiting value of the parameter we have finitely many periodic orbits (for each fixed numerator in the rotation number), passed this value there is a sudden birth of an infinite number of them. We find that these two bifurcations are remarkably connected forming the main “skeleton” of the global bifurcation diagram. We conjecture that this type of global connection must be present in related problems such as the classic isosceles three-body problem and the anisotropic Kepler problem.

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