Pseudo-heteroclinic connections between bicircular restricted four-body problems

In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems –BR4BP –(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed.Peer ReviewedPostprint (published version

The Restricted 3-Body Problem on S1 : regularization and a particular solution

We study a special case of the two body problem when the particles are restricted to move in the space S1. We regularize all collisions using a symplectic transformation and classify the trajectories in four families. After that, we add a third infinitesimal body, getting four restricted three body problems on S1 corresponding to each one of the previous families. Then, we apply only one symplectic transformation that regularizes all the singularities due to binary collisions between the infinitesimal body with each primary. We show the global dynamics in one of the restricted problems, when the primaries are fixed at the poles of S1. We exhibit a particular set of solutions which takes place when the primaries perform hyperbolic motions only

Research in the Restricted Problems of Three and Four Bodies Final Scientific Report

Seven studies have been conducted on research in the existence and nature of solutions of the restricted problems of three and four bodies. The details and results of five of these research investigations have already been published, and the latest two studies will be published shortly. A complete bibliography of publications is included in this report. This research has been primarily qualitative and has yielded new information on the behavior of trajectories near the libration points in the Earth-Moon-Sun and Sun-Jupiter-Saturn systems, and on the existence of periodic trajectories about the libration points of the circular and elliptical restricted four-body models. We have also implemented Birkhoff's normalization process for conservative and nonconservative Hamiltonian systems with equilibrium points. This makes available a technique for analyzing stability properties of certain nonlinear dynamical systems, and we have applied this technique to the circular and elliptical restricted three-body models. A related study was also conducted to determine the feasibility of using cislunar periodic trajectories for various space missions. Preliminary results suggest that this concept is attractive for space flight safety operations in cislunar space. Results of this research will be of interest to mathematicians, particularly those working in ordinary differential equations, dynamical systems and celestial mechanics; to astronomers; and to space guidance and mission analysts

Symmetries and Symmetry Breaking

In understanding the world of matter, the introduction of symmetry principles following experimentation or using the predictive power of symmetry principles to guide experimentation is most profound. The conservation of energy, linear momentum, angular momentum, charge, and CPT involve fundamental symmetries. All other conservation laws are valid within a restricted subspace of the four interactions: the strong, the electromagnetic, the weak, and the gravitational interaction. In this paper comments are made regarding parity violation in hadronic systems, charge symmetry breaking in two nucleon and few nucleon systems, and time-reversal-invariance in hadronic systems.Comment: 5 Pages, LaTeX, 2 PostScript figures. Talk at 17th International IUPAP Conference on Few-body Problems in Physics, June 5-10, 2003, Durham, North Carolina, US

A Heuristic Strategy to Compute Ensemble of Trajectories for 3D Low Cost Earth-Moon Transfers

The problem of finding optimal trajectories is essential for modern space mission design. When considering multibody gravitational dynamics and exploiting both low-thrust and high-thrust and alternative forms of propulsion such as solar sailing, sets of good initial guesses are fundamental for the convergence to local or global optimal solutions, using both direct or indirect methods available to solve the optimal control problem. This paper deals with obtaining preliminary trajectories that are designed to be good initial guesses as input to search optimal low-energy short-time Earth-Moon transfers with ballistic capture. A more realistic modelling is introduced, in which the restricted four-body system Sun-Earth-Moon-Spacecraft is decoupled in two patched planar Circular Restricted Three-Body Problems, taking into account the inclination of the orbital plane of the Moon with respect to the ecliptic. We present a heuristic strategy based on the hyperbolic invariant manifolds of the Lyapunov orbits around the Lagrangian points of the Earth- Moon system to obtain ballistic capture orbits around the Moon that fulfill specific mission requirements. Moreover, quasi-periodic orbits of the Sun-Earth system are exploited using a genetic algorithm to find optimal solutions with respect to total Dv, time of flight and altitude at departure. Finally, the procedure is illustrated and the full transfer trajectories assessed in view of relevant properties. The proposed methodology provides sets of low-cost and shorttime initial guesses to serve as inputs to compute fully optimized three-dimensional solutions considering different propulsion technologies, such as low, high, and hybrid thrust, and/or using more realistic models

On the Advantages of Using a Strict Hierarchy to Model Astrodynamical Problems

In this paper an algorithm is developed that combines the capabilities and advantages of several different astrodynamical models of increasing complexity. Splitting these models in a strict hierarchical order yields a clearer grasp on what is available. With the effort of developing a comprehensive model overhead, the equations for the spacecraft motion in simpler models can be readily obtained as particular cases. The proposed algorithm embeds the circular and elliptic restricted three-body problems, the four-body bicircular and concentric models, an averaged n-body model, and, at the top hierarchic ladder, the full ephemeris spice-based restricted n-body problem. The equations of motion are reduced to the assignment of 13 time-varying coefficients, which multiply the states and the gravitational potential to reproduce the proper vector field. This approach is powerful because it allows, for instance, an efficient and quick way to check solutions for different dynamics and parameters. It is shown how a gradual increase of the dynamics complexity greatly improves accuracy, the chances of success and the convergence rate of a continuation algorithm, applied to low-energy transfers

On large deformations of thin elasto-plastic shells: Implementation of a finite rotation model for quadrilateral shell element

A large-deformation model for thin shells composed of elasto-plastic material is presented in this work, Formulation of the shell model, equivalent to the two-dimensional Cosserat continuum, is developed from the three-dimensional continuum by employing standard assumptions on the distribution of the displacement held in the shell body, A model for thin shells is obtained by an approximation of terms describing the shell geometry. Finite rotations of the director field are described by a rotation vector formulation. An elasto-plastic constitutive model is developed based on the von Mises yield criterion and isotropic hardening. In this work, attention is restricted to problems where strains remain small allowing for all aspects of material identification and associated computational treatment, developed for small-strain elastoplastic models, to be transferred easily to the present elasto-plastic thin-shell model. A finite element formulation is based on the four-noded isoparametric element. A particular attention is devoted to the consistent linearization of the shell kinematics and elasto-plastic material model, in order to achieve quadratic rate of asymptotic convergence typical for the Newton-Raphson-based solution procedures. To illustrate the main objective of the present approach-namely the simulation of failures of thin elastoplastic shells typically associated with buckling-type instabilities and/or bending-dominated shell problems resulting in formation of plastic hinges-several numerical examples are presented, Numerical results are compared with the available experimental results and representative numerical simulations

Studies on restricted suckling in dual purpose and dairy breed cattle in Mexico

The aim of this thesis was to investigate the effects of rearing calves by restricted suckling (RS) compared to artificial rearing (AR) in dual purpose and dairy breed cattle in Mexico, milked once or three times a day, respectively. The following parameters were recorded during the first eight weeks after calving: social, abnormal and general behaviours and weight gain of Zebu crossbred and dairy calves; and milk yield, milk composition and udder health in the Zebu crossbred dams and udder health and milk let-down in the Holstein dams. RS reduced the abnormal behaviours of the calves and resulted in less foraging behaviours compared to AR. AR dairy calves consumed four-fold as much concentrate as RS dairy calves, whereas the amount of milk ingested was similar in the two treatments. A higher fat content in the milk ingested by RS calves compared to whole milk fed to AR calves, resulted in similar total ME intake from milk and concentrate in both treatments. Irrespective of type of animals RS resulted in similar weight gain in comparison to AR. Cow-calf separation five days after calving induced fewer indications of physiological stress in RS cows and calves. RS further increased the saleable and total (saleable and suckled) milk yield and decreased the fat content in saleable milk. RS improved udder health of the Zebu crossbred cows as judged according to elevated CMT scores and lower lactose content in AR cows. RS of Holstein dairy cows did not impair milk let-down and udder health tended to be improved according to the CMT. RS calves showed a front teat preference, which did not impair udder health according to similar CMT scores of front and rear teats. These studies indicate that RS is economically viable for the farmer as once daily milking combined with twice daily suckling considerably increased the saleable and total milk yield of Zebu crossbred cows and improved animal well-being