Some counterexamples to a generalized Saari's conjecture

For the Newtonian -body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.

     

Minima de l'intégrale d'action et équilibres relatifs de n corps

We consider the n-body problem in an Euclidean space E of finite dimension for a homogeneous potential which is invariant under isometries, in particular the Newtonian potential. We show that the absolute minima of the action integral among zero mean loops of class H¹ whose period is fixed coincide with the relative equilibrium solutions whose configuration is an absolute minimum of the potential among the configurations whose moment of inertia with respect to their center of mass is fixed. The result is at the same time general and elementary and does not appeal to functional analysis.     

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.     

Saari's conjecture is true for generic vector fields

The simplest non-collision solutions of the -body problem are the ``relative equilibria', in which each body follows a circular orbit around the centre of mass and the shape formed by the bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian -body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel, Trans. Amer. Math. Soc. (2005). We present a different kind of answer: proofs that several generalisations of Saari's conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.

     

The problem of the time-optimal turning of a manipulator

Two types of manipulator that perform three-dimensional motions are considered, and the control problem in which the manipulator rotation is performed in minimum time is studied. The rate of rotation of a rigid body about an axis rises as the moment of inertia about this axis falls. Manipulator control amounts to a problem of the rotation of a system of rigid bodies about an axis. In addition to the angle of rotation, there is a further controlled coordinate, whose variation can vary the moment of inertia about the axis. Assuming that the moment of inertia can be stantaneously “frozen” (that pulse control signals are possible), the in-time-optimal control modes were found in /1, 2/, (see also Akulenko, L.D. et al., “Optimization of the control modes of manipulation robots”, Preprint 218, In-t. Problem Mekhaniki Akad. Nauk SSSR, Moscow 1983). In these modes, the rotation, occurs in the entire time interval with minimum moment of inertia about the axis of rotation. The rotation when there are constraints on the control (pulse control signals are not permitted) was considered in /3/. Numerical studies there led to the false conclusion that, in the optimal motion, with a finite number of control switchings, the moment of inertia is also a minimum throughout the time interval. Below, for a set of extreme configurations, a control is constructed for the two types of manipulator, which satisfies the Pontryagin maximum principle, when there are constraints on the control signals. During its rotation the manipulator section then performs oscillations about a position corresponding to minimum moment of inertia about the axis of rotation. It is shown that the motion considered in /3/, which contains a singular mode with minimum moment of inertia, is not optimal. The motion which satisfies the maximum principle is compared with it. There can be a singular mode in the optimal motion /4/ only when the number of control switchings is infinite.     

Bifurcations of the relative equilibria of a gyrostat satellite for a special case of the alignment of its gyrostatic moment

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.     

Topology optimization of space vehicle structures considering attitude control effort

The topology optimization of load-bearing structural components for reducing attitude control efforts of miniature space vehicles is investigated. Based on the derivation of the cold gas consumption rate of three-axis stabilization actuators, it is pointed out that the attitude control efforts associated with cold gas micro thrusters are closely related to the mass moment inertia of the system. Therefore, the need to restrict the mass moments of inertia of the structural components is highlighted in the design of the load-bearing structural components when the attitude control performance is concerned. The optimal layout design of the space vehicle structure considering attitude control effort is, thus, reformulated as a topology optimization problem for minimum compliance under constraints on mass moments of inertia. Numerical techniques for the optimization problem are discussed. For the case of a single constraint on the mass moment of inertia about a given axis, a design variable updating scheme based on the Karush–Kuhn–Tucker optimality criteria is used to solve the minimization problem. For the problem with multiple constraints, mathematical programming approach is employed to seek the optimum. Numerical examples will be given to demonstrate the validity and applicability of the present problem statement.     

Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2

We analyze the two‐dimensional parabolic‐elliptic Patlak‐Keller‐Segel model in the whole Euclidean space ?². Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local‐in‐time existence for any mass of “free‐energy solutions,” namely weak solutions with some free‐energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free‐energy solutions with initial data as before for the critical mass 8π/χ. Actually, we prove that solutions blow up as a delta Dirac at the center of mass when t → ∞ when their second moment is kept constant at any time. Furthermore, all moments larger than 2 blowup as t → ∞ if initially bounded. © 2007 Wiley Periodicals, Inc.     

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.     

Properties of solutions for the generalized Euler‐Poisson equations

The present paper deals with equations, which generalize the known Euler‐Poisson equations for the motion of a heavy rigid body about a fixed point. These equations arise in dynamics of systems of coupled rigid bodies. In these equations the generalized inertia tensor depends upon components of vertical vector, i.e. it is not constant. Our aim is to analyze Lyapunov stability of stationary solutions and orbital stability of periodic solutions of the equations under study.     

Saari's conjecture for the collinear -body problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian -body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

     

The global stability of the steady rotations of a solid

The dynamical Euler equations describing the motion of a non-symmetrical solid about the centre of mass in the field of a constant external moment and a dissipative one are considered. It is assumed that the external moment specified with respect to axes attached to the body acts about the intermediate central axis of inertia of the body. The conditions for global asymptotic stability as well as the stability in total of steady rotations of the solid are obtained.     

Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations

The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time-shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.     

Relative Equilibria in the Spherical,Finite Density Three-Body Problem

The relative equilibria for the spherical, finite density three-body problem are identified. Specifically, there are 28 distinct relative equilibria in this problem which include the classical five relative equilibria for the point-mass three-body problem. None of the identified relative equilibria exist or are stable over all values of angular momentum. The stability and bifurcation pathways of these relative equilibria are mapped out as the angular momentum of the system is increased. This is done under the assumption that they have equal and constant densities and that the entire system rotates about its maximum moment of inertia. The transition to finite density greatly increases the number of relative equilibria in the three-body problem and ensures that minimum energy configurations exist for all values of angular momentum.     

Optimal control of the rotation of a system of two bodies connected by an elastic rod

Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved.     

Global Solutions to the Coupled Chemotaxis-Fluid Equations

In this paper, we are concerned with a model arising from biology, which is a coupled system of the chemotaxis equations and the viscous incompressible fluid equations through transport and external forcing. The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the Chemotaxis-Navier–Stokes system over three space dimensions, we obtain global existence and rates of convergence on classical solutions near constant states. When the fluid motion is described by the simpler Stokes equations, we prove global existence of weak solutions in two space dimensions for cell density with finite mass, first-order spatial moment and entropy provided that the external forcing is weak or the substrate concentration is small.     

Determination of the parameters of a rigid body clamped at an end of a beam from the natural frequencies of vibrations

The unique reconstruction is proven of the parameters of a rigid body (mass, moment of inertia, and static moment of inertia) clamped at one of the ends of the Timoshenko beam from the natural frequencies of its vibrations. We suggest a new method of reconstruction and exhibit an example.     

Single-axis equilibrium orientations to the attracting center of a symmetrical prolate orbital gyrostat with an elastic rod

Under study in the restricted formulation is the motion of a symmetrical prolate stationary gyrostat along a Keplerian circular orbit in a central Newtonian field of forces. An elastic homogeneous rod, rectilinear in the undeformed state, is rigidly clamped by one end in the body of gyrostat along its axis of symmetry. There is a point mass at the free end of the rod. The inextensible elastic rod, for simplicity of constant circular cross-section, performs infinitesimal space oscillations in the process of system motion. In this case, we neglect the terms in the system’s tensor of inertia which are nonlinear with respect to displacements of the points of the rod.We consider the following (so-called semi-inverse) problem: Under what kinetic momentumof the flywheel, among the relative equilibriums of the system (the states of rest relative to the orbital coordinate system) does there exist an equilibrium such that the axis, arbitrarily chosen in the coordinate system associated with the gyrostat, is collinear with the local vertical? In the discretization of the problem, we present the values of the Lagrange coordinates that define the deformation of the rod for these equilibria and the value of gyrostatic moment providing the presence of the equilibrium in question.     

On the Application of the Perturbation Method in the Buckling Optimization of a Heavy Elastic Column of Constant Aspect Ratio

This paper addresses the problem of maximizing the first Euler load of a column of constant aspect ratio (moment of inertia varies with the square of the cross-sectional area), considering the column self-weight, subject to the constraint of fixed volume of column material and height. The coupled nonlinear integro-differential equations of optimality and stability, generated through variational principles, have been solved using the method of parameter perturbation. The optimal column has cross-sectional area that follows a perturbed two-thirds power law; the first Euler load is up to 87% larger than that of the corresponding column of uniform cross-section having the same volume and height.