scholarly journals Is Jacobi theorem valid in the singly averaged restricted circular Three-Body-Problem?

2021 ◽  
Vol 8 (1) ◽  
pp. 179-184
Author(s):  
Konstantin V. Kholshevnikov ◽  
◽  
Keyword(s):  
Total Energy ◽  
Body Problem ◽  
Negative Energy ◽  
Mass Point ◽  
Restricted Three Body Problem ◽  
Energy Integral ◽  
Three Body Problem ◽  
Zero Mass ◽  
N Body Problem ◽  
Three Body

C. Jacobi found that in the General N-Body-Problem (including N = 3) for the Lagrangian stability of any solution necessary is the negativity of the total energy of the system. For the restricted three-body-problem, this statement is trivial, since a zero-mass body introduces zero contribution to the energy of the system. If we consider only the equations describing the movement of the zero mass point, then the energy integral disappears. However, if we average the equations over the longitudes of the main bodies, the energy integral appears again. Is the Jacobi theorem valid in this case? It turned out not. For arbutrary large values of total energy, there exist bounded periodic orbits. At the same time the negative energy is sufficient for the boundedness of an orbit in the configuration space.

scholarly journals On the Evection Resonance and Its Connection to the Stability of Outer Satellites

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Tadashi Yokoyama ◽  
Ernesto Vieira Neto ◽  
Othon Cabo Winter ◽  
Diogo Merguizo Sanchez ◽  
Pedro Ivo de Oliveira Brasil
Keyword(s):  
Body Problem ◽  
Semimajor Axis ◽  
Simplified Model ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Strong Instability ◽  
Analytical Derivation ◽  
N Body Problem ◽  
The Stability ◽  
Three Body

In terms of stability around the primary, it is widely known that the semimajor axis of the retrograde satellites is much larger than the corresponding semimajor axis of the prograde satellites. Usually this conclusion is obtained numerically, since precise analytical derivation is far from being easy, especially, in the case of two or more disturbers. Following the seminal idea that what is unstable in the restricted three-body problem is also unstable in the general N-body problem, we present a simplified model which allows us to derive interesting resonant configurations. These configurations are responsible for cumulative perturbations which can give birth to strong instability that may cause the ejection of the satellite. Then we obtain, analytically, approximate bounds of the stability of prograde and retrograde satellites. Although we recover quite well previous results of other authors, we comment very briefly some weakness of these bounds.

On the modified circular restricted three-body problem with variable mass

New Astronomy ◽  
2021 ◽  
Vol 84 ◽  
pp. 101510
Author(s):  
Md Sanam Suraj ◽  
Rajiv Aggarwal ◽  
Md Chand Asique ◽  
Amit Mittal
Keyword(s):  
Body Problem ◽  
Variable Mass ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

scholarly journals GEOMETRY OF HOMOCLINIC CONNECTIONS IN A PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1151-1169 ◽  
Author(s):  
MARIAN GIDEA ◽  
JOSEP J. MASDEMONT
Keyword(s):  
Symbolic Dynamics ◽  
Body Problem ◽  
Invariant Manifolds ◽  
Libration Point ◽  
Homoclinic Orbits ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lyapunov Orbits ◽  
Homoclinic Connections ◽  
Three Body

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1and the other collinear libration points L2, L3is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries.

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