Families of planar orbits generated by homogeneous potentials

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f _y /f _x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.     

On the adelphic potentials compatible with a set of planar orbits

Given a planar potentialB=B(x, y), compatible with a monoparametric family of planar orbitsf(x, y)=c, we face the problem of producing potentialsA=A(x, y), adelphic toB(x, y), i.e. nontrivial potentials which have in common withB(x, y) the given set of orbits. We establish a linear, second order partial differential equation for a functionP(x, y) and we prove that, to any definite positive solution of this equation, there corresponds a potentialA(x, y) adelphic toB(x, y).     

Boundary curves for families of planar orbits

For monoparametric familiesf(x,y)=c of planar orbits, created by a planar potentialV(x,y), we introduce the notion of the family boundary curves (FBC). All members of the familyf(x,y)=c are traced in an allowable region of thexy plane, defined by the corresponding FBC, with total energyE=E(c) varying along the family. Family boundary curves are also found for two-parametric familiesf(x,y,b)=c. The relation of equilibrium points and asymptotic orbits, possibly possessed by the potentialV(x,y), to be FBC is studied.     

Separable and partially separable systems in the light of the inverse problem of dynamics

Using a generalization of Joukovsky's formula, we determine three-dimensional families of curves that are orbits only in separable potentials and we note the importance of iso-energetic families of orbits. We also obtain more general families that are orbits of partially separable systems and we examine from this point of view the classical curvilinear coordinate systems.     

Continuation of normal doubly symmetric orbits in conservative reversible systems

In this paper we introduce the concept of a quasi-submersive mapping between two finite-dimensional spaces, we obtain the main properties of such mappings, and we introduce “normality conditions” under which a particular class of so-called “constrained mappings” are quasi-submersive at their zeros. Our main application is concerned with the continuation properties of normal doubly symmetric orbits in time-reversible systems with one or more first integrals. As examples we study the continuation of the figure-eight and the supereight choreographies in the N-body problem.     

Inhomogeneous potentials producing homogeneous orbits

We prove that, in general, a given two-dimensional inhomogeneous potential V(x,y) does not allow for the creation of homogeneous families of orbits. Yet, depending on the case at hand, if the given potential satisfies certain conditions, this potential is compatible either with one (or two) monoparametric homogeneous families of orbits or at most with five such familes. The orbits are then found on the grounds of the given potential.     

On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = U _i(q ¹, q ²,..., q ⁿ)q ⁱ + U(q ¹, q ²,...;q ⁿ) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U ₁, U ₂,...;, U _n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M ₂ we obtain, for potential function U(q ¹, q ²), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.     

Stability of axial orbits in galactic potentials

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.     

On the construction of dynamic systems from given integrals

In this communication we propose a new approach for studying a particular type of inverse problems in mechanics related to the construction of a force field from given integrals.An extension of the Danielli problem is obtained. The given results are applied to the Suslov problem, and illustrated in specific examples.     

Stability of axial orbits in analytical galactic potentials

We study the stability of axial orbits in analytical galactic potentials as a function of the energy of the orbit and the ellipticity of the potential. The problem is solved by an analytical method, the validity of which is not limited to small amplitudes. The lines of neutral stability divide the parameter space in regions corresponding to different organizations of the main families of orbits in the symmetry planes.     

Periodic orbits and escapes in dynamical systems

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central ??bumpy?? black hole. When the energy reaches its escape value, a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.     

Periodic orbits in three-dimensional planetary systems

Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5 and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of the vertical critical orbits of the corresponding planar problem     

Periodic orbits for a class of galactic potentials

In this work, applying general results from averaging theory, we find periodic orbits for a class of Hamiltonian systems H whose potential models the motion of elliptic galaxies.     

Keplerian periodic orbits in the isosceles problem

The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision. The basic tool used is the study of the Poincaré maps associated with the periodic solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of planetary orbits in binary systems

The possible existence of stable orbits is investigated in binary systems using Hill's method. Analytical stability conditions are established for satellites, for inner planets and for outer planets, allowing arbitrary values for the mass-ratio of the binary.Presented at the Symposium Star Catalogues, Positional Astronomy and Celestial Mechanics, held in honor of Paul Herget at the U.S. Naval Observatory, Washington, November 30, 1978.     

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60^° of mutual planetary inclination, but in most families, the stability does not exceed 20^°–30^°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.     

Two-parametric families of orbits created by three-dimensional galactic-type potentials

Astrophysics and Space Science - Fast radio bursts (FRBs) are extremely strong radio flares lasting several micro- to milliseconds and come from unidentified objects at cosmological distances, most...     

On periodic orbits and resonance in extrasolar planetary systems

We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω ₂ ? ω ₁) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω ₂ ? ω ₁) librates or rotates are not different, but are closely connected to the exact periodic motion.     

Impulsive time-free transfers between halo orbits

A methodology is developed to design optimal time-free impulsive transfers between three-dimensional halo orbits in the vicinity of the interior L ₁ libration point of the Sun-Earth/Moon barycenter system. The transfer trajectories are optimal in the sense that the total characteristic velocity required to implement the transfer exhibits a local minimum. Criteria are established whereby the implementation of a coast in the initial orbit, a coast in the final orbit, or dual coasts accomplishes a reduction in fuel expenditure. The optimality of a reference two-impulse transfer can be determined by examining the slope at the endpoints of a plot of the magnitude of the primer vector on the reference trajectory. If the initial and final slopes of the primer magnitude are zero, the transfer trajectory is optimal; otherwise, the execution of coasts is warranted. The optimal time of flight on the time-free transfer, and consequently, the departure and arrival locations on the halo orbits are determined by the unconstrained minimization of a function of two variables using a multivariable search technique. Results indicate that the cost can be substantially diminished by the allowance for coasts in the initial and final libration-point orbits.An earlier version was presented as Paper AIAA 92-4580 at the AIAA/AAS Astrodynamics Conference, Hilton Head Island, SC, U.S.A., August 10–12, 1992.