The equations of motion of planar elliptic restricted three body problem are transformed to four decoupled Hill's equations. By using the Floquet theorem analytic solution to the oscillator equations with time dependent periodic coefficients are presented. We show that the new analytic approach is valid for system parameters $0 < e \leq 0.05$ and $0 < μ\leq 0.01$ where $e$ denotes the eccentricity of primaries while $μ$ is the mass parameter, respectively. We also clarify the transformation details that provide the applicability of the method.
Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects. This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries. The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hami…
Post-Newtonian effects on the stability of the triangular solution in the three-body problem for general masses
在早期出版物[Ichita，Yamada and Asada，Phys. Rev. D { \bf 83}，084026（2011）; Yamada and Asada，Phys. Rev. D { \bf 86}，124029（2012）]中开始的继续工作，我们研究了后牛顿（PN）对一般质量的相对论性三体问题中三角解稳定性的影响。对于三个有限质量，三角形解的稳定性条件得到的第一后牛顿（1 PN）的顺序，它恢复以前的结果PN限制三体问题时，一个质量趋于零。即使在1 PN阶下，稳定区域仍然存在，尽管对于一般质量的PN三角形构型不如PN限制的三体情形以及牛顿情形稳定。
The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of $μ, h,σ_1$ and $σ_2$ ($h$ is energy constant, $μ$ mass ratio of the two primaries, $σ_1$ and $σ_2$ are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky \cite{Kari}. We have applied the predictor-corrector algorithm to construct the periodic orbits in an…
Triangular solution to general relativistic three-body problem for general masses
在早期出版物[Ichita，Yamada和Asada，Phys. Rev. D 83，084026（2011）]中开始的继续工作，我们重新研究了后牛顿对三体问题拉格朗日等边三角形解的影响。对于三个有限质量的三角形位形，当且仅当它的每一个边长都有相对论修正时，它满足广义相对论中的后牛顿运动方程.这个后牛顿配置为三个有限的质量并不总是等边的，它恢复了以前的结果限制三体问题时，一个质量为零。对于相同的质量和角速度，后牛顿三角形构型总是小于牛顿三角形构型。
In the last decades a peculiar family of solutions of the Circular Restricted Three Body Problem has been used to explain the temporary captures of small bodies and spacecrafts by a planet of the Solar System. These solutions, which transit close to the Lagrangian points $L_1,L_2$ of the CRTBP, have been classified using the values of approximate local integrals and of the Jacobi constant. The use for small bodies of the Solar System requires to consider a hierarchical extension of the model, from the CRTBP to the the full $N$ planetary problem. The Elliptic Restricted Three Body, which is the…
In the preliminary design of space missions it can be useful to identify regions of dynamics that drive the system's behaviour or separate qualitatively different dynamics. The Lagrangian Coherent Structure (LCS) has been widely used in the analysis of dynamical systems, and generalises the concept of the stable and unstable manifolds to systems with arbitrary time-dependence. However, the use of three-dimensional LCS in astrodynamics has thus far been limited. This paper presents the application of a new numerical method introduced by the authors, DA-LCS, to astrodynamics systems using the El…
The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1-L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-…
