On the generalized restricted problem of three bodies

1974 ◽  
Vol 22 ◽  
pp. 85
Author(s):  
G. N. Duboshin
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Active Material ◽  
The Body ◽  
The Other ◽  
Three Body Problem ◽  
Straight Line ◽  
Second Derivatives ◽  
Material Points ◽  
Three Body

AbstractThe particular case of the complete generalized three-body problem where one of the body-points does not exert influence on the other two is analysed. These active material points act on the passive point and also on each other with forces (attraction or repulsion), proportional to the product of masses of both points and a certain function of the time, their mutual distances and their first and second derivatives. Furthermore it is not supposed that generally the th ird axiom of mechanics (action = reaction) takes place.One determines the conditions for some particular solutions which exist, when the three points form the equilateral triangle or remain always on a straight line.Finally, some conclusions on the Liapunov stability in the simplest cases are drawn.

scholarly journals New Phenomena in the Spatial Isosceles Three-Body Problem

2015 ◽  
Vol 25 (09) ◽  
pp. 1550116 ◽  
Author(s):  
Duokui Yan ◽  
Tiancheng Ouyang
Keyword(s):  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Initial Condition ◽  
Collision Orbit ◽  
Three Body Problem ◽  
Straight Line ◽  
Three Body ◽  
New Phenomena ◽  
Two Bodies

In the three-body problem, it is known that there exists a special set of periodic orbits: spatial isosceles periodic orbits. In each period, one body moves up and down along a straight line, and the other two bodies rotate around this line. In this work, we revisit this set of orbits by applying variational method. Two unexpected phenomena are discovered. First, this set is not always spatial. It actually bifurcates from the circular Euler (central configuration) orbit to the Broucke (collision) orbit. Second, one of the orbits in this set encounters an oscillating behavior. By running its initial condition, the orbit stays periodic for only a few periods before it becomes irregular. However, it moves close to another periodic shape in a while. Shortly it falls apart again and starts running close to a third periodic shape after a moment. This oscillation continues as t increases. Actually, up to t = 1.2 × 105, the orbit is bounded and keeps oscillating between periodic shapes and irregular motions.

scholarly journals Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem

1999 ◽  
Vol 172 ◽  
pp. 445-446 ◽  
Author(s):  
Giancarlo Benettin ◽  
Francesco Fassò ◽  
Massimiliano Guzzo
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Critical Mass ◽  
Kam Theory ◽  
Birkhoff Normal Form ◽  
Three Body Problem ◽  
Arnold Diffusion ◽  
Convexity Assumption ◽  
Three Body ◽  
Nekhoroshev Stability

The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).

scholarly journals A Complete Family of Periodic Solutions of the Planar Three-Body Problem and their Stability

1977 ◽  
Vol 33 ◽  
pp. 159-159
Author(s):  
M. Hénon
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Three Body Problem ◽  
Complete Family ◽  
First Order ◽  
Hierarchical Arrangement ◽  
The Family ◽  
Three Body

AbstractWe give a complete description of a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with a rectilinear orbit, computed by Schubart in 1956. It ends in retrograde revolution, i.e., a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the “interplay” type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.

Extended Camus Theory and Higher Order Conjugated Curves

2019 ◽  
Author(s):  
Cody Leeheng Chan ◽  
Kwun-Lon Ting
Keyword(s):  
Stationary Point ◽  
Higher Order ◽  
The Body ◽  
The Other ◽  
Body System ◽  
Straight Line ◽  
Pair Generation ◽  
Curvature Theory ◽  
Three Body ◽  
Two Bodies

Abstract According to Camus’ theorem, for a single DOF 3-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold-Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e. slide along itself), on the line of the instant centers a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotate about a stationary point, the stationary point embedded on the body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of the Camus’ theorem. The Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.

Displacement of the Lagrange Equilibrium Points in the Restricted Three Body Problem With Rigid Body Satellite

1978 ◽  
Vol 41 ◽  
pp. 305-314
Author(s):  
W.J. Robinson
Keyword(s):  
Rigid Body ◽  
Restricted Problem ◽  
Body Problem ◽  
Equilibrium Points ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

AbstractIn the restricted problem of three point masses, the positions of the equilibrium points are well known and are tabulated. When the satellite is a rigid body, these values no longer correspond to the equilibrium points. This paper seeks to determine the magnitudes of the discrepancies.

scholarly journals Quasi-Lntegrals of the Plane Restricted Three-Body Problem

1993 ◽  
Vol 132 ◽  
pp. 309-319
Author(s):  
E.M. Nezhinskij
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Test Body ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Jacobi Integral ◽  
Three Body ◽  
Two Bodies

AbstractThe paper is concerned with studying the domain of possible motion and a field of the test body velocities in the plane restricted problem of three bodies. The study is based on existence of a quasi-integral of areas (similar to an integral of areas in the problem of two bodies) as well as on the Jacobi integral. The method of constructing the quasi-integrals is a standard one (see, for example, [1],[2].

Resonant Three-Dimensional Periodic Solutions About the Triangular Equilibrium Points in the Restricted Problem

1983 ◽  
Vol 74 ◽  
pp. 235-247 ◽  
Author(s):  
C.G. Zagouras ◽  
V.V. Markellos
Keyword(s):  
Periodic Solutions ◽  
Restricted Problem ◽  
Body Problem ◽  
Equilibrium Points ◽  
Mass Parameter ◽  
Three Dimensional ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Special Values ◽  
Three Body

AbstractIn the three-dimensional restricted three-body problem, the existence of resonant periodic solutions about L4 is shown and expansions for them are constructed for special values of the mass parameter, by means of a perturbation method. These solutions form a second family of periodic orbits bifurcating from the triangular equilibrium point. This bifurcation is the evolution, as μ varies continuously, of a regular vertical bifurcation point on the corresponding family of planar periodic solutions emanating from L4.

A third order canonical approximation of the general solution of the restricted problem of three bodies in the neighbourhood of L4

1966 ◽  
Vol 25 ◽  
pp. 170-175
Author(s):  
A. Deprit
Keyword(s):  
General Solution ◽  
Restricted Problem ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Order ◽  
The Third ◽  
Transformation Of Variables ◽  
Definition Of ◽  
Three Body

A canonical transformation of variables is introduced in the plane restricted three-body problem which gives the Hamiltonian in the form of a power series with normalized second order terms. Then a generating function is constructed, step by step, that permits the definition of new action and angle variables, such that the Hamiltonian is independent of the angle variables. This procedure has been done explicitly up to the third order terms.

Special cases of the restricted problem of three bodies

1966 ◽  
Vol 25 ◽  
pp. 187-193 ◽  
Author(s):  
J. Schubart
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Disturbing Function ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Long Period ◽  
Period Effects ◽  
Special Cases ◽  
Three Body

The long-period effects in nearly commensurable cases of the restricted three-body problem were studied according to the ideas of Poincaré. The secular and critical terms of the disturbing function were isolated by a numerical averaging process, by use of an IBM 7094 computer.

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