Using variational minimizing methods,we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.
Truncating at the second order of the mutual potential between two rigid bodies,time-explicit rst order solutions to the rotations and the orbital motion of the two bodies in the planar full two-body problem(F2BP)are ...Truncating at the second order of the mutual potential between two rigid bodies,time-explicit rst order solutions to the rotations and the orbital motion of the two bodies in the planar full two-body problem(F2BP)are constructed.Based on this analytical solution,equations of motion(EOMs)for the related restricted full three-body problem are given.In the case of the synchronous or double synchronous states for the full two-body problem,EOMs for the related restricted full three-body problem(RF3BP)are also given.At last,one example-the"collinear libration point"in the binary asteroid system-is given.展开更多
In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R^3 or R^2 are precisely the planar equilateral triangle circular solutions found by J. Lagran...In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R^3 or R^2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular rostricted 3-body problem in R^3 or R^2 are also planar equilateral triangle circular solutions.展开更多
For any given positive masses it is proved that the variational minimization solutions of the 3-body problem in 3 or 2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and...For any given positive masses it is proved that the variational minimization solutions of the 3-body problem in 3 or 2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in 3 or 2 are also planar equilateral triangle circular solutions.展开更多
We study the charged 3-body problem with the potential function being （-a）-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the m...We study the charged 3-body problem with the potential function being （-a）-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the τ/2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.展开更多
Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is t...Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）. In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.展开更多
This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails.By including the time-varying sail orientation in the linearized equations of motion for the circular restr...This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails.By including the time-varying sail orientation in the linearized equations of motion for the circular restricted three-body problem(CR3BP),four types of quasi-periodic orbits(two types around L1 and two types around L2)were formulated.Among them,one type of orbit around L2 realizes a considerably small geometry variation while ensuring visibility from the Earth if(and only if)the sail acceleration due to solar radiation pressure is approximately of a certain magnitude,which is much smaller than that assumed in several previous studies.This means that only small solar sails can remain in the vicinity of L2 for a long time without propellant consumption.The orbits designed in the linearized CR3BP can be translated into nonlinear CR3BP and high-fidelity ephemeris models without losing geometrical characteristics.In this study,new quasi-periodic orbits are formulated,and their characteristics are discussed.Furthermore,their extendibility to higher-fidelity dynamic models was verified using numerical examples.展开更多
Stable or nearly stable orbits do not generally possess well-distinguished manifold structures that assist in designing trajectories for departing from or arriving onto a periodic orbit.For some potential missions,the...Stable or nearly stable orbits do not generally possess well-distinguished manifold structures that assist in designing trajectories for departing from or arriving onto a periodic orbit.For some potential missions,the orbits of interest are selected as nearly stable to reduce the possibility of rapid departure.However,the linearly stable nature of these orbits is also a drawback for their timely insertion into or departure from the orbit.Stable or nearly stable near rectilinear halo orbits(NRHOs),distant retrograde orbits(DROs),and lunar orbits offer potential long-horizon trajectories for exploration missions and demand eficient operations.The current investigation focuses on leveraging stretching directions as a tool for departure and trajectory design applications.The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and,therefore,offers information for efficient departure from the orbit.Similarly,maximum stretching in reverse time enables arrival with a minimal maneuver magnitude.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11071175)a grant for advisor and PhD students from educational committee of China
文摘Using variational minimizing methods,we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.
基金National Natural Science Foundation of China(Grant Nos.11322330 and 11673072)National Basic Research Program of China(Grant No.2013CB834100).
文摘Truncating at the second order of the mutual potential between two rigid bodies,time-explicit rst order solutions to the rotations and the orbital motion of the two bodies in the planar full two-body problem(F2BP)are constructed.Based on this analytical solution,equations of motion(EOMs)for the related restricted full three-body problem are given.In the case of the synchronous or double synchronous states for the full two-body problem,EOMs for the related restricted full three-body problem(RF3BP)are also given.At last,one example-the"collinear libration point"in the binary asteroid system-is given.
基金Partially supported by the NNSF and MCME of China. the Qiu Shi Sci. and Tech. Foundation.Edn. Comm. of Tianjun CityAssociate Member of the ICTP.Partially supported by the NNSF of China
文摘In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R^3 or R^2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular rostricted 3-body problem in R^3 or R^2 are also planar equilateral triangle circular solutions.
文摘For any given positive masses it is proved that the variational minimization solutions of the 3-body problem in 3 or 2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular restricted 3-body problem in 3 or 2 are also planar equilateral triangle circular solutions.
基金The authors thank sincerely Professor Shanzhong Sun for his careful reading and helpful comments on the manuscript of this paper. The first author was partially supported by the Doctoral Innovation Project of Nankai University. The second author was partially supported by the National Natural Science Foundation of China （Grant No. 11131004）, MCME, LPMC of Ministry of Education of China, Nankai University, and the BCMIIS at Capital Normal University.
文摘We study the charged 3-body problem with the potential function being （-a）-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the τ/2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.
文摘Following Jacobi＇s geometrization of Lagrange＇s least action principle, trajectories of classical mechanics can be characterized as geodesics on the configuration space M with respect to a suitable metric which is the conformal modification of the kinematic metric by the factor （U ＋ h）, where U and h are the potential function and the total energy, respectively. In the special case of 3-body motions with zero angular momentum, the global geometry of such trajectories can be reduced to that of their moduli curves, which record the change of size and shape, in the moduli space of oriented m-triangles, whose kinematic metric is, in fact, a Riemannian cone over the shape space M^＊≌S^2 （1/2）. In this paper, it is shown that the moduli curve of such a motion is uniquely determined by its shape curve （which only records the change of shape） in the case of h≠0, while in the special case of h = 0 it is uniquely determined up to scaling. Thus, the study of the global geometry of such motions can be further reduced to that of the shape curves, which are time-parametrized curves on the 2-sphere characterized by a third order ODE. Moreover, these curves have two remarkable properties, namely the uniqueness of parametrization and the monotonieity, that constitute a solid foundation for a systematic study of their global geometry and naturally lead to the formulation of some pertinent problems.
文摘This paper proposes new quasi-periodic orbits around Earth–Moon collinear libration points using solar sails.By including the time-varying sail orientation in the linearized equations of motion for the circular restricted three-body problem(CR3BP),four types of quasi-periodic orbits(two types around L1 and two types around L2)were formulated.Among them,one type of orbit around L2 realizes a considerably small geometry variation while ensuring visibility from the Earth if(and only if)the sail acceleration due to solar radiation pressure is approximately of a certain magnitude,which is much smaller than that assumed in several previous studies.This means that only small solar sails can remain in the vicinity of L2 for a long time without propellant consumption.The orbits designed in the linearized CR3BP can be translated into nonlinear CR3BP and high-fidelity ephemeris models without losing geometrical characteristics.In this study,new quasi-periodic orbits are formulated,and their characteristics are discussed.Furthermore,their extendibility to higher-fidelity dynamic models was verified using numerical examples.
文摘Stable or nearly stable orbits do not generally possess well-distinguished manifold structures that assist in designing trajectories for departing from or arriving onto a periodic orbit.For some potential missions,the orbits of interest are selected as nearly stable to reduce the possibility of rapid departure.However,the linearly stable nature of these orbits is also a drawback for their timely insertion into or departure from the orbit.Stable or nearly stable near rectilinear halo orbits(NRHOs),distant retrograde orbits(DROs),and lunar orbits offer potential long-horizon trajectories for exploration missions and demand eficient operations.The current investigation focuses on leveraging stretching directions as a tool for departure and trajectory design applications.The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and,therefore,offers information for efficient departure from the orbit.Similarly,maximum stretching in reverse time enables arrival with a minimal maneuver magnitude.
