On normal coordinates in the vicinity of the Lagrangian libration points of the restricted elliptic three-body problem

A planar restricted elliptic three-body problem is considered. The motions close to the triangular libration points are studied. The problem parameters (the eccentricity of the orbit of the main attracting bodies and the ratio of their masses) are assumed to lie inside the linear stability region of the libration points. The magnitude of eccentricity is considered small. A linear canonical, periodic in true anomaly transformation is obtained analytically up to the second degree of eccentricity inclusive that reduces the Hamiltonian function of the linearized equations of perturbed motion to real normal form in the vicinity of the libration points. This form corresponds to two harmonic oscillators not connected to one another, with frequencies depending on the problem parameters. In constructing the normalizing canonical transformation, the Depri-Hori method of the perturbation theory of Hamiltonian systems is used. Its implementation in the problem under study relies heavily on computer systems of analytical calculations.

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A Note on a Separation of Equations of Variation of the Elliptic Restricted Three-Body Problem Into Hill's Equations

Symposium - International Astronomical Union ◽

10.1017/s0074180900070480 ◽

1974 ◽

Vol 62 ◽

pp. 131-131

Author(s):

V. Matas

Keyword(s):

Body Problem ◽

Libration Points ◽

True Anomaly ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Triangular Libration Points ◽

Straight Line ◽

Hill’S Equations ◽

Image Position ◽

Three Body

The equations of variation of the three-dimensional elliptic restricted three-body problem corresponding to the equilibrium solutions (the libration points) have been separated into three Hill's equations. As regards the equation ‘corresponding’ to the motion of the infinitesemal body in the z-axis (perpendicular to the plane of motion of the primaries), the matter is trivial one since the initial equation - as known - reads d2z/dv2 + (Ai + e cosv)/(l + e cosv) = 0 (e, 0<e< 1, and v are the eccentricity and the true anomaly of the relative motion of the primaries) with Ai > 1 for the straight-line libration points Li (i= 1, 2, 3) and Ai=l for the triangular libration points Li, i=4, 5. As concerns the remaining two components, x and y, of the motion of the infinitesimal body (x, y and z are the Nechvíle's variables), in the case of the straight-line libration points, L1, L2 and L3, the corresponding equations of variation have been transformed and separated into two further - mutually independent - Hill's equations without any limitation. In the case of the equilateral triangle libration points, L4 and L5, the separation has been found only when the eccentricity e and the dimensionless mass μ, 0<μ≦1/2 of the ‘minor’ primary satisfy the additional conditions: Let us write the latter two Hill's equations obtained in the form where Ik, k= 1, 2, are 2π-periodic even functions of the true anomaly v. The functions Ik, k = 1, 2, are real functions in the case of the straight-line libration points, L1, L2 and L3, without a limitation but in the case of the triangular libration points, L4 and L5, they are real only if Provided the functions Ik, k= 1, 2, are complex-valued functions of the real variable v.

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