Heterogeneous primaries in CR4BP

This paper investigates the motion of the massless body moving under the influence of the gravitational forces of the three equal heterogeneous oblate spheroids placed at Lagrangian configuration. After determining the equations of motion and the Jacobian constant of the massless body, we have illustrated the numerical work (Stationary points, zero-velocity curves, regions of motion, Poincare surfaces of section and basins of attraction). And then we have checked the linear stability of the stationary points and found that all the stationary points are unstable.

The Hénon and Heiles Problem in Three Dimensions.

This paper is the first part of a study of the Hénon and Heiles problem in three dimensions. Due to the axial symmetry of the Hamiltonian, the third component of the angular momentum is an integral and the system is considered as a Hamiltonian with two degrees of freedom. As functions of that integral, we show the existence of three circular trajectories around the axis Oz and a domain for which we have bounded motions. In the part of that domain near the origin, the corresponding dynamical system is treated as a perturbed harmonic oscillator in 1–1–1 resonance. We present some numerical studies searching for periodic orbits, showing the corresponding Poincaré surfaces of section. In addition, we obtain some natural families of periodic orbits associated with the relative equilibria of the fourth order normalized system.

ACTION INTEGRALS AND ENERGY SURFACES OF THE KOVALEVSKAYA TOP

The different types of energy surfaces are identified for the Kovalevskaya problem of rigid body dynamics, on the basis of a bifurcation analysis of Poincaré surfaces of section. The organization of their foliation by invariant tori is qualitatively described in terms of Poincaré-Fomenko stacks. The individual tori are then analyzed for sets of independent closed paths, using a new algorithm based on Arnold’s proof of the Liouville theorem. Once these paths are found, the action integrals can be calculated. Energy surfaces are constructed in the space of action variables, for six characteristic values of energy. The data are presented in terms of color graphs that give an intuitive access to this highly complex integrable system.

Deterministic Chaos in the Dynamics of a Freely Jointed Chain with Three Degrees of Freedom

The dynamics of a short freely jointed chain of three segments is investigated numerically. The chain consists of mass points connected by massless rigid rods, its initial and final points being fixed. Thus the chain represents a holonomically constrained system with three degrees of freedom. It is shown that the motion of the mass points can be chaotic; the occurrence of chaos depends on the initial conditions of the motion, the end-to-end distance of the chain, and the angular momentum about the axis of the stretching direction. Moreover, the chain more likely exhibits regular than chaotic behavior. The numerical results are presented in the form of Poincare surfaces of section, including the use of a slice technique, as well as in the form of power spectra.