K. E. Papadakis

EQUILIBRIUM POINTS AND THEIR STABILITY IN THE RESTRICTED FOUR-BODY PROBLEM

We study numerically the problem of four bodies, three of which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle, while the fourth is infinitesimal. The fourth body does not affect the motion of the three bodies (primaries). The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves as well as the positions of the equilibrium points are given. The existence and the number of collinear and noncollinear equilibrium points of the problem depend on the mass parameters of the primaries. For three unequal masses, collinear equilibrium solutions do not exist. Critical masses associated with the existence and the number of equilibrium points, are given. The stability of the relative equilibrium solutions in all cases is also studied. The regions of the basins of attraction for the equilibrium points of the present dynamical model for some values of the mass parameters are illustrated.

NON-LINEAR STABILITY ZONES AROUND TRIANGULAR EQUILIBRIA IN THE PLANE CIRCULAR RESTRICTED THREE-BODY PROBLEM WITH OBLATENESS

Non-linear stability zones of the triangular Lagrangian points are computed numerically in the case of oblate larger primary in the plane circular restricted three-body problem. It is found that oblateness has a noticeable effect and this is identified to be related to the resonant cases and the associated curves in the mass parameter μ versus oblateness coefficientA1 parameter space.

The 3D restricted three-body problem under angular velocity variation

Nonlinear approximation of periodic motions around the collinear equilibrium points in the case of the restricted three-body problem when the angular velocity of the primaries is not equal to the value of the classical problem (which is unit in the usual units of mass, length and time), is studied. The stability of the equilibrium points and the analytical solutions in their neighborhood constructing series approximations of the periodic orbits in the planar and in the spatial problem, are given. Families which emanate from L1, L2 and L3 both in the plane and in three dimensions as well as their stability for the Earth-Moon mass distribution, are computed. Special generating plane orbits, the vertical-critical orbits, of the families a, b and c of the problem, are determined and presented. We have also computed series of vertical-critical periodic orbits with the angular velocity as parameter. Three-dimensional families which generate from the bifurcation orbits for ω = 2, are given.

Application of the Characteristic Bisection Method for locating and computing periodic orbits in molecular systems

The Characteristic Bisection Method for finding the roots of non-linear algebraic and/or transcendental equations is applied to LiNC/LiCN molecular system to locate periodic orbits and to construct the continuation/bifurcation diagram of the bend mode family. The algorithm is based on the Characteristic Polyhedra which define a domain in phase space where the topological degree is not zero. The results are compared with previous calculations obtained by the Newton Multiple Shooting algorithm. The Characteristic Bisection Method not only reproduces the old results, but also, locates new symmetric and asymmetric families of periodic orbits of high multiplicity.  2001 Elsevier Science B.V. All rights reserved.

Families of periodic orbits in the photogravitational three-body problem

All the families of planar symmetric simple-periodic orbits of the photogravitational restricted plane circular three-body problem, are determined numerically in the case when the primaries are of equal mass and radiate with equal radiation factors (q1=q2=q). We obtain a global view of the possible patterns of periodic three-body motion while the full range of values of the common radiation factor is explored, from the gravitational case (q=1) down to near the critical value at which the triangular equilibria disappear by coalescing with the inner equilibrium pointL1 on the rotating axis of the primaries. It is found that for large deviations of its value from the gravitational case the radiation factorq can have a strong effect on the structure of the families.

The stability of inner collinear equilibrium points in the photogravitational elliptic restricted problem

The linear stability of the inner collinear equilibrium point of the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity and radiation pressure. The case of equal radiation factors of the two primaries is considered and the full range of values of the common radiation factor is explored, from the caseq1 =q2 =q = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries transferring their stability to the collinear point, down toq = 0 at which value the stability regions in theµ - e plane disappear by shrinking down to zero size. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case. They evolve as in the case of the triangular equilibrium point considered in a previous paper. There exist values of the common radiation factor, in the range considered, for which the collinear equilibrium point is stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.

A new simple method for the analytical solution of Kepler's equation

A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules.

Non-Linear Stability Zones Around the Triangular Lagrangian Points

Non-linear stability zones of the triangular Lagrangian points are determined numerically and the effect of the parameters of mass distribution and eccentricity of primaries are considered within the framework of the elliptic restricted three-body problem. It is found that both parameters have a strong effect reducing the stability zones to negligible size for some parameter values within the linear stability regions. The effect is identified to be due to the non-linearly unstable resonant cases and the associated curves in the μ-e parameter space. It is thus concluded that the classical μ-e linear stability diagram is only partially valid in terms of the more practical concept of non-linear stability.

THE PLANAR PHOTOGRAVITATIONAL HILL PROBLEM

In the planar photogravitational Hill problem (PHP) three equivalent forms of the equations of motion are presented: the physical equations of motion and two more forms of the regularized equations of the problem. Using the Levi–Civita coordinate transformation as well as the corresponding time transformation, we first obtain a simple regularized polynomial Hamiltonian of the dynamical system, and second the regularized equations of motion without using a regularized Hamiltonian. The relation between the physical and the regularized variables are also given. Zero velocity curves and surfaces, Poincare sections, families of symmetric periodic orbits as well as their stability are given in physical and regularized variables when the larger primary body (the sun) does not radiate and when it does. It is found that for large deviations of its value from the gravitational case, the radiation factor can have a strong effect on the structure of the families. Finally we present the stability regions in both cases with and without radiation using the maximal Lyapunov exponent (LE), working in the regularized equations of motion of the problem since through these the calculation of the stability is more rapid and accurate than using the physical equations of motion.