Reduction,relative equilibria and potential in the two rigid bodies problem

In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies.     

Effects of the Triaxiality on the Rotation of Celestial Bodies: Application to the Earth,Mars and Eros

In this paper we discuss the influence of the triaxiality of a celestialbody on its free rotation, i.e. in absence of any external gravitationalperturbation. We compare the results obtained through two different analytical formalisms, one established from Andoyer variables by usingHamiltonian theory, the other one from Euler's variables by usingLagrangian equations. We also give a very accurate formulation of thepolar motion (polhody) in the case of a small amplitude of this motion.Then, we carry out a numerical integration of the problem, with aRunge–Kutta–Felberg algorithm, and for the two kinds of methods above, that we apply to three different celestial bodies considered as rigid : the Earth, Mars, and Eros. The reason of this choice is that each of this body corresponds to a more or less triaxial shape.In the case of the Earth and Mars we show the good agreement betweenanalytical and numerical determinations of the polar motion, and theamplitude of the effect related to the triaxial shape of the body, whichis far from being negligible, with some influence on the polhody of theorder of 10 cm for the Earth, and 1 m for Mars. In the case of Eros, weuse recent output data given by the NEAR probe, to determine in detailthe nature of its free rotational motion, characterized by the presence ofimportant oscillations for the Euler angles due to the particularly largetriaxial shape of the asteroid.     

The equilibrium configurations of the restricted problem of 2+2 triaxial rigid bodies

The restricted 2+2 body problem is considered. The infinitesimal masses are replaced by triaxial rigid bodies and the equations of motion are derived in Lagrange form. Subsequently, the equilibrium solutions for the rotational and translational motion of the bodies are detected. These solutions are conveniently classified in groups according to the several combinations which are possible between the translational equilibria and the constant orientations of the bodies.     

A global regularisation of the gravitationalN-body problem

The work of Aarseth and Zare (1974) is extended to provide aglobal regularisation of the classical gravitational three-body problem: by transformation of the variables in a way that does not depend on the particular configuration, we obtain equations of motion which are regular with respect to collisions between any pair of particles. The only cases excepted are those in which collisions between more than one pair occur simultaneously and those in which at least one of the masses vanishes. However, by means of the same principles the restricted problem is regularised globally if collisions between the two primaries are excluded. Results of numerical tests are summarised, and the theory is generalised to provide global regularisations, first, for perturbed three-body motion and, second, for theN-body problem. A way of increasing the number of degrees of freedom of a dynamical system is central to the method, and is the subject of an Appendix.     

Short-axis-mode rotation of a free rigid body by perturbation series

A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter.     

Elements of spin motion

For use in numerical studies of rotational motion, a set of elements is introduced for the torque-free rotational motion of a rigid body around its barycenter. The elements are defined as the initial values of a modification of the Andoyer canonical variables. A computational procedure is obtained for determining these elements from the combination of the spin angular momentum vector and a triad defining the orientation of the rigid body. A numerical experiment shows that the errors of transformation between the elements and variables are sufficiently small. The errors increase linearly with time for some elements and quadratically for some others.     

Attitude dynamics of a rigid body on a Keplerian orbit: A simplification

An infinitestimal contact transformation is proposed to simplify at first order the Hamiltonian representing the attitude of a triaxial rigid body on a Keplerian orbit around a mass point. The simplified problem reduces to the Euler-Poinsot model, but with moments of inertia depending on time through the longitude in orbit. Should the orbit be circular, the moments of inertia would be constant.     

On the roto-translatory motion of a satellite of an oblate primary

We deal with the problem of the motion of a triaxial satellite of an oblate primary of larger mass. We show that the treatment is simplified by using a canonical set of variables previously introduced by the authors, that allows a drastic reduction in the expansions of the potential. A general method to avoid the appearance of virtual singularities when the angles between certain planes are small is designed. Our approach is applicable either to natural or artificial satellites.     

On the attitude dynamics of perturbed triaxial rigid bodies

Attitude dynamics of perturbed triaxial rigid bodies is a rather involved problem, due to the presence of elliptic functions even in the Euler equations for the free rotation of a triaxial rigid body. With the solution of the Euler–Poinsot problem, that will be taken as the unperturbed part, we expand the perturbation in Fourier series, which coefficients are rational functions of the Jacobian nome. These series converge very fast, and thus, with only few terms a good approximation is obtained. Once the expansion is performed, it is possible to apply to it a Lie-transformation. An application to a tri-axial rigid body moving in a Keplerian orbit is made.     

Equivalent problems in rigid body dynamics — I

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.     

Behavior of nearby synchronous rotations of a Poincaré–Hough satellite at low eccentricity

This paper presents a study of the Poincaré–Hough model of rotation of the synchronous natural satellites, in which these bodies are assumed to be composed of a rigid mantle and a triaxial cavity filled with inviscid fluid of constant uniform density and vorticity. In considering an Io-like body on a low eccentricity orbit, we describe the different possible behaviors of the system, depending on the size, polar flattening and shape of the core. We use for that the numerical tool. We propagate numerically the Hamilton equations of the system, before expressing the resulting variables under a quasi-periodic representation. This expression is obtained numerically by frequency analysis. This allows us to characterise the equilibria of the system, and to distinguish the causes of their time variations. We show that, even without orbital eccentricity, the system can have complex behaviors, in particular when the core is highly flattened. In such a case, the polar motion is forced by several degrees and longitudinal librations appear. This is due to splitting of the equilibrium position of the polar motion. We also get a shift of the obliquity when the polar flattening of the core is small.     

Out-of-plane equilibrium points and their stability in the Robe’s problem with oblateness and triaxiality

This paper examines the existence and stability of the out-of-plane equilibrium points of a third body of infinitesimal mass when the equations of motion are written in the three dimensional form under the set up of the Robe’s circular restricted three-body problem, in which the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second one is a triaxial rigid body under the full buoyancy force of the fluid. The existence of the out of orbital plane equilibrium points lying on the xz-plane is noticed. These points are however unstable in the linear sense.     

Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem

This paper studies the motion of an infinitesimal mass around triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. A practical application of this case could be the study of motion of a satellite under the effect of Sun and Earth. We have exploited the method of averaging used by Grebnikov (Nauka, Moscow, revised 1986) throughout the analysis of stability of the system. The critical mass ratio depends on the radiation pressure, oblateness, eccentricity and semi major axis of the elliptic orbits and the range of stability decreases as the radiation parameter increases.     

Exact linearization of non-planar intermediary orbits in the satellite theory

In this paper we consider the reduction of the equations of motion for non-planar perturbed two body problems into linear form. It is seen that this can be easily accomplished for any element of the class of radial intermediaries to the satellite problem proposed by Deprit in 1981, since they have a functional dependence suitable for linearization. The transformation is worked out by using an adequate set of redundant variables. Four harmonic oscillators are obtained, of which two are coupled through gyroscopic terms. Their constant frequencies contain the secular contribution of the main problem of artificial satellite theory up to the order of the considered intermediary. Therefore, this result may well be interesting in relation to the study and prediction of accurate long-term solutions to satellite problems.     

Interacting ellipsoids: a minimal model for the dynamics of rubble-pile bodies

A simple mechanical model is formulated to study the dynamics of rubble-pile asteroids, formed by the gravitational re-accumulation of fragments after the collisional breakup of a parent body. In this model, a rubble-pile consists of N interacting fragments represented by rigid ellipsoids, and the equations of motion explicitly incorporate the minimal degrees of freedom necessary to describe the attitude and rotational state of each fragment. In spite of its simplicity, our numerical examples indicate that the overall behavior of our model is in line with several known properties of collisional events, like the energy and angular momentum partition during high velocity impacts. Therefore, it may be considered as a well defined minimal model.     

Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque

The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to arbitrary initial angular velocity. In the paper a parametrization of the rotation by three complex numbers is used. In particular, the rows of the rotation matrix are seen as elements of the unit sphere and projected, by stereographic projection, onto points on the complex plane. In this representation, the kinematic differential equation reduces to an equation of Riccati type, which is solved through appropriate choices of substitutions, thereby yielding an analytic solution in terms of confluent hypergeometric functions. The rotation matrix is recovered from the three complex rotation variables by inverse stereographic map. The results of a numerical experiment confirming the exactness of the analytic solution are reported. The newly found analytic solution is valid for any motion time length and rotation amplitude. The present paper adds a further element to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

A note on the application of a generalized canonical approach to non-linear Hamiltonian systems

In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only.     

A reduction of the magnetic-binary problem by means of the energy integral

The author's aim in this article has been to present the magnetic-binary problem in a new form different from the usual, by using a process of reduction of a dynamical problem to another one with fewer degrees of freedom.     

The Hamiltonian Dynamics of the Two Gyrostats Problem

The problem of two gyrostats in a central force field is considered. We prove that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system posseses symmetries. Using them we perform the reduction of the number of degrees of freedom. We show that at every stage of the reduction process, equations of motion are Hamiltonian and give explicit forms corresponding to non-canonical Poisson brackets. Finally, we study the case where one of the gyrostats has null gyrostatic momentum and we study the zero and the second order approximation, showing that all equilibria are unstable in the zero order approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability of the classical type of relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem is generalized to that of a rigid body in a J ₂ gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J ₂ and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.