The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

Linear stability of relative equilibria in the charged three-body problem

A relative equilibrium is a periodic orbit of the n-body problem that rotates uniformly maintaining the same central configuration for all time. In this paper we generalize some results of R. Moeckel and we apply it to study the linear stability of relative equilibria in the charged three-body problem. We find necessary conditions to have relative equilibria linearly stable for the collinear charged three-body problem, for planar relative equilibria we obtain necessary and sufficient conditions for linear stability in terms of the parameters, masses and electrostatic charges. In the last case we obtain a stability inequality which generalizes the Routh condition of celestial mechanics. We also proof the existence of spatial relative equilibria and the existence of planar relative equilibria of any triangular shape.     

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-body system is discussed for the scheme presented in this paper first.     

Isolating blocks near the collinear relative equilibria of the three-body problem

The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.

     

Dynamics in the Schwarzschild Isosceles Three Body Problem

The Schwarzschild potential, defined as \(U(r)=-A/r-B/r^3\) , where \(r\) is the relative distance between two mass points and \(A,B>0\) , models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy–momentum diagram. We further employ appropriate regularization transformations to analyze the behavior of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for nonzero total angular momenta (e.g., when two of the mass points spin infinitely many times around the center of mass). This phenomenon is known in celestial mechanics as the black-hole effect and is understood as an analog in the classical context of behavior near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits that are neither homothetic nor homographic.     

Stability Analysis of an Orbiting Plate with Finite Elements

This paper contributes to the analysis of rotating equilibria of non–driven systems (sometimes called relative equilibria), which are characterized by the fact that their angular momentum is conserved and non–zero. Interesting applications are usually found in space dynamics, in particular if large earth orbiting structures such as the space elevator are considered. For flexible structures relative equilibria can be found with Finite Element software packages. However, in this case the stability analysis is a non–trivial task since one usually has only limited account to the internal data of commercial FE solvers. Therefore, in [5] a new finite element based stability test was developed and applied to one–dimensional structural elements. Here, we consider a large–scale flexible plate orbiting around the earth in order to demonstrate that this method works well also for two dimensional shell–elements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and differential geometric properties of continuous families of periodic three-body motions with non-uniform mass distributions

Using the method of analytic continuation in an equivariant differential geometric setting, we exhibit two interesting families of vanishing angular momentum periodic orbits for the Newtonian three-body problem with non-uniform mass distributions having two equal masses which connect at the celebrated figure-8 orbit, exhibited by A. Chenciner and R. Montgomery (2000) in the case of equal masses, and yield a continuous family of periodic three-body motions in the plane.At one end of the family, when the two equal masses are infinitesimal and the third one reaches the value of +1, we arrive at a solution of a double Kepler problem; at the other end of the family, when the third mass is infinitesimal, we have a special case of periodic solution of a restricted three-body problem.     

Adiabatic representation in the Coulomb three-body problem in the united-atom limit: Nuclear widths of the energy levels of the muonic molecule <Emphasis Type="Italic">ttµ</Emphasis>

We study the asymptotic behavior of the wave function of the system of three Coulomb particles in the united-atom limit in the adiabatic representation of the three-body problem. This result is used to calculate the nuclear widths of muonic-molecule energy levels. We discuss features of the approach with regard to excited states of the muonic molecule ttµ with a nonzero orbital angular momentum.     

Three-dimensional manifestly Poincaré-invariant approach to the relativistic three-body problem

A three-dimensional manifestly Poincaré-invariant approach to the relativistic three-body problem is developed that satisfies the requirement of cluster separability and at the same time does not lead to so-called spurious states devoid of physical meaning. It is shown that these requirements make it possible to fix the form of the operators of the two-body interactions. The problem is solved with allowance for the dependence of the interaction operators on the spectral parameter. This dependence is a manifestation of the structure of the particles in the three-body system (i.e., it reflects the circumstance that the complete Hilbert space of state vectors of the system includes not only three-body configurations of the original particles) and leads to the appearance of certain factors in the cross sections of physical processes. Two alternative formulations of the method are investigated. In the first formulation, equations are written down for the amplitudes of transitions between free-particle states. In the second formulation, the states of interacting particles in the two-body scattering channels are used as complete orthogonal bases. Partial-wave expansions of the equations with respect to states with given total angular momentum of the system in the helicity basis are made.Institute of Nuclear Physics of the State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 200–232, May, 1995.     

From the restricted to the full three-body problem

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.     

The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.     

Relative Equilibria of Molecules

Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0 . Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY ₂ , X ₃ , and XY ₄ molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G -invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual {\frak g} ^* and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in {\frak g} ^* . Received June 9, 1997; second revision received December 15, 1997; final revision received January 19, 1998     

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.     

Unitary rotation of square-pixellated images

The unitary rotation of square-pixellated images is based on the finite su(2)-oscillator model, which describes systems whose values for position, momentum and energy, are discrete and finite. In a two-dimensional position space, this allows the construction of angular momentum states, orthonormal and complete, for which rotations are defined as multiplication by phases that carry the rotation angle. The decomposition of a digital square images in terms of these angular momentum states determines a unitary (hence invertible) rotation of the image, whose kernel can be computed as a four-dimensional array of real numbers.     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (? 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 ?λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

The properties of the steady motions of a body carrying a system of two-degree-of-freedom powered gyroscopes

The steady motions of a rigid body carrying several two-degree-of-freedom powered gyroscopes in a uniform external field are investigated. It is shown that when the installation scheme of the gyroscopes in the carrying body is collinear, the problem of determining the steady motions of the system and analysing their secular stability reduces for the most part to the previously solved, similar problem for a system with one gyroscope. It is established that when there is dissipation in the axes of the gyroscope frames, the system tends asymptotically to a state of rest if the absolute value of the total angular momentum of the system lies in the segment of possible absolute values of the angular momentum of the gyroscope rotors. The results of an analysis of the steady motions of a system carrying two gyroscopes with a non-collinear installation scheme are presented.     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

On the Bifurcation and Stability of Rigidly Rotating Inviscid Liquid Bridges

Summary. We consider a mathematical model that describes the motion of an ideal fluid of finite volume that forms a bridge between two fixed parallel plates. Most importantly, this model includes capillarity effects at the plates and surface tension at the free surface of the liquid bridge. We point out that the liquid can stick to the plates due to the inner pressure even in the absence of adhesion forces. We use both the Hamiltonian structure and the symmetry group of this model to perform a bifurcation and stability analysis for relative equilibrium solutions. Starting from rigidly rotating, circularly cylindrical fluid bridges, which exist for arbitrary values of the angular velocity and vanishing adhesion forces, we find various symmetry-breaking bifurcations and prove corresponding stability results. Either the angular velocity or the angular momentum can be used as a bifurcation parameter. This analysis reduces to find critical points and corresponding definiteness properties of a potential function involving the respective bifurcation parameter. Received June 21, 1996; revision received October 2, 1997, and accepted for publication October 9, 1997     

A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

     

Asymptotic Analysis of Linear Feedback Nash Equilibria in Nonzero-Sum Linear-Quadratic Differential Games

In this paper, we discuss nonzero-sum linear-quadratic differential games. For this kind of games, the Nash equilibria for different kinds of information structures were first studied by Starr and Ho. Most of the literature on the topic of nonzero-sum linear-quadratic differential games is concerned with games of fixed, finite duration; i.e., games are studied over a finite time horizon t _f. In this paper, we study the behavior of feedback Nash equilibria for t _f.In the case of memoryless perfect-state information, we study the so-called feedback Nash equilibrium. Contrary to the open-loop case, we note that the coupled Riccati equations for the feedback Nash equilibrium are inherently nonlinear. Therefore, we limit the dynamic analysis to the scalar case. For the special case that all parameters are scalar, a detailed dynamical analysis is given for the quadratic system of coupled Riccati equations. We show that the asymptotic behavior of the solutions of the Riccati equations depends strongly on the specified terminal values. Finally, we show that, although the feedback Nash equilibrium over any fixed finite horizon is generically unique, there can exist several different feedback Nash equilibria in stationary strategies for the infinite-horizon problem, even when we restrict our attention to Nash equilibria that are stable in the dynamical sense.