Andoyer Variables Research Articles

The Serret-Andoyer formalism in rigid-body dynamics: I. Symmetries and perturbations

This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, which stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into the action-angle ones, using the method of Sadov.

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).

The Rotation of Europa

We present a semi-analytical theory of the rotation of Europa the Galilean satellite of Jupiter. The theory is semi-analytical in the sense that it is based on a synthetic theory of the orbit of Europa developed by Lainey. The theory is developed in the framework of Hamiltonian mechanics, using Andoyer variables and assumes that Europa is a rigid body. We consider this theory as a first step toward the modelization of a non rigid Europa covered by an ocean.

Spacecraft’s attitude prediction: solar radiation torque and the Earth’s shadow

Abstract A semi-analytical approach is proposed to study the rotational motion of an artificial satellite, under the influence of torque due to the solar radiation pressure, and taking into account the influence of Earth’s shadow. Using Andoyer variables the equations for the rotational motion are presented in extended Hamiltonian form. In order to get a solution for the state variables close to an actual motion, the considered model for the shadow function takes into account physical and geometric factors and three specific regions: shadow, penumbra and full light. A mapping for the shadow function is proposed and a semi-analytical process is applied. When the satellite is totally illuminated or it is inside the penumbra, a known analytical solution is used to compute the satellite’s attitude. A numerical simulation shows, when the penumbra region is included, the attenuation of the rotational motion during the transition from the shadow to the illuminate region and vice versa.

Rotation of Synchronous Satellites Application to the Galilean Satellites

An analytical theory of the rotation of a synchronous satellite is developed for the application to the rotation of the Galilean satellites. The theory is developed in the framework of Hamiltonian mechanics, using Andoyer variables. Special attention is given to the frequencies of libration as functions of the moments of inertia of the satellite.

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 celestial body on its free rotation, i.e. in absence of any external gravitational perturbation. We compare the results obtained through two different analytical formalisms, one established from Andoyer variables by using Hamiltonian theory, the other one from Euler's variables by using Lagrangian equations. We also give a very accurate formulation of the polar motion (polhody) in the case of a small amplitude of this motion. Then, we carry out a numerical integration of the problem, with a Runge-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 between analytical and numerical determinations of the polar motion, and the amplitude of the effect related to the triaxial shape of the body, which is far from being negligible, with some influence on the polhody of the order of 10 cm for the Earth, and 1 m for Mars. In the case of Eros, we use recent output data given by the NEAR probe, to determine in detail the nature of its free rotational motion, characterized by the presence of important oscillations for the Euler angles due to the particularly large triaxial shape of the asteroid.

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.

Evolution of Rotational Motion of a Symmetrical Satellite with Viscoelastic Rods

We study the motion of a symmetrical satellite with a pair of flexible viscoelastic rods in a central Newtonian gravitational field. A restricted problem formulation is considered, when the satellite's center of mass moves along a fixed circular orbit. A small parameter e is introduced which is inversely proportional to the stiffness of flexible elements. Another small parameter μ is equal to the ratio of the squared orbital angular velocity and the squared magnitude of the initial angular velocity of the satellite. In order to describe the satellite rotational motion relative to the center of mass, we use the canonical Andoyer variables. In the undisturbed formulation of the problem, i.e., at e = 0 and μ = 0, these variables are the action–angle variables. Equations describing the evolution of motion are derived by an asymptotic method which combines the method of separating motions for systems with an infinite number of degrees of freedom and the Krylov–Bogolyubov method for systems with fast and slow variables. The manifolds of stationary motions are found, and their stability is investigated on the basis of equations in variations. Phase portraits are constructed which describe the rotational motion of a satellite at the stage of slow dissipative evolution.

Evolution of Motion of a Binary Planet

A model of a binary planet, consisting of a material point of small mass and a deformable viscoelastic sphere, is suggested. The center of mass of the binary planet moves in the gravitational field of a central body in the plane, which contains planets forming the binary planet. A deformable spherical planet rotates around the axis orthogonal to the plane of planetary motion. Planet deformations are described by the linear theory of viscoelasticity. It is shown that with an appropriate approximation of the gravitational potential, there is a class of quasicircular orbits, when the eccentricities of an orbit of the center of mass of a binary planet and an orbit, describing mutual planet motion, are equal to zero. The further evolution of motion is investigated in this class of orbits with the use of the canonical Poincare–Andoyer variables. Corresponding averaged equations are found, and phase pictures are constructed; the stability of stationary solutions is investigated on the basis of equations in variations. For the Solar system planets with their satellites, forming binary planets, the application of the presented model allows us to conclude that satellites sooner or later will fall on the corresponding planets.

On integrable cases of the poincare problem

The equations of motion for the classical Poincare problem of the rotational motion of a rigid body with an ellipsoidal cavity containing liquid in canonical Andoyer variables have been obtain. Thr...

Perturbated rotational motion of weakly deformable celestial bodies

Abstract The rotation equations of weakly deformable celestial bodies (in canonical and non-canonical Andoyer variables) are developed in detail. A theory of the perturbed rotational motion of an isolated weakly deformable body has been developed. Applications to Earth's rotation theory are given.

Tidal Variations of the Earth Rotation

AbstractThe equations for the rotation of a weakly deformable celestial body in non canonical Andoyer variables have been used to study the perturbation of Earth rotation due to tidal deformation raised by the Moon and Sun. A theory of the perturbed rotational motion of an isolated weakly deformable body in Andoyer variables and in components of the angular velocity has been developed. Mantle tidal deformations due to lunar and solar influences were analytically described and taken into account. Perturbations of the first order in the Earth’s polar motion were determined.

The evolution of a rotational motion of an elastic- rigid body about a fixed point in a gravitational field

The rotational motion of a rigid dynamical system including a three dimensional elastic part about a fixed point is studied. The system is under the action of a general gravitational field. The stability anlysis is carried out by expanding the displacement of the deformable part in terms of orthonormalized functions with the independent coefficients. Results are obtained easily based on Routh's equation and Andoyer variables. The system is found to be stable provided that the dissipation energy is minimum corresponds to the case of axially symmetric deformation. On the other hand, the system is unstable, when the angular momentum vector is parallel to the equatorial plane.

Unperturbed chandler motion and perturbation theory of the rotation motion of deformable celestial bodies

Abstract New unperturbed motions are suggested for the study of the rotational motion of deformable celestial bodies. This motion describes the rotation of an isolated celestial body deformed by its own rotation. By some natural simplifications and by using special forms of canonical variables (similar to Andoyer's variables) the problem is reduced to the classical Euler-Poinsot problem for a rigid body, but with different moments of inertia. The suggested unpertubed motion describes Chandler's pole motion and we shall call it Chandler or Euler-Chandler motion. The development of the unperturbed theory is described in this paper. The solution of the Chandler problem (Andoyer's variables, components of angular velocity of the body's axes, and their direction cosines) is presented in elliptical and θ- functions, and in the form of Fourier series in the angle-action variables. Similar Fourier series were obtained for products and squares of the diraction cosines. The coefficients of these series are expresse...

Influence of Earth's shadow on the rotational motion of an artificial satellite perturbed by solar radiation torque

A semi-analytical approach is proposed to study the rotational motion of an artificial satellite under the influence of the torque due to the solar radiation pressure and taking into account the influence of Earth's shadow. The Earth's shadow is introduced in the equations for the rotational motion as a function depending on the longitude of the Sun, on the ecliptic's obliquity and on the orbital parameters of the satellite. By mapping and computing this function, we can get the periods in which the satellite is not illuminated and the torque due to the solar radiation pressure is zero. When the satellite is illuminated, a known analytical solution is used to predict the satellite's attitude. This analytical solution is expressed in terms of Andoyer's variables and depends on the physical and geometrical properties of the satellite and on the direction of the Sun radiation flux. By simulating a hypothetical circular cylindrical type satellite, an example is exhibited and the results agree quite well when compared with a numerical integration.

Kinetic energy of a non-spherical elastic earth mantle with andoyer variables

Continuing the study of the rotation of a deformable Earth begun by Getino and Ferrandiz (1990, 1991a, 1991b, 1993, 1994) for an Earth model with an elastic spherical mantle, in this paper on one hand we deal with the effect of the ellipticity, and on the other hand, we include the toroidal solution of the displacement vector. Taking an axis symmetrical, slightly ellipsoidal Earth, the modification due to the ellipticity is introduced into the solution of the displacement vector for both spheroidal and toroidal modes, and, after defining the adequate variables, we give the canonical formulation of the corresponding increase in the kinetic energy.

On the motion around the centre of mass of a viscously elastic sphere in the restricted elliptic three-body problem

In the restricted three-body problem we consider the motion of a viscously elastic sphere (planet) with its centre of mass moving in a conditionally-periodic orbit. The approximate equations describing the rotational motion of the sphere in terms of the Andoyer variables are obtained by the method of the separation of motion and averaging; the evolution of the motion is also analysed.

Rotational evolution of a symmetric gyrostat with visco-elastic bars around the center of mass in a circular orbit

The motion of a gyrostat in a circular orbit in a Newtonian field of force is considered. The gyrostat has four homogeneous viscoelastic bars attached to it. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Bending deformations of the bars, accompanied by dissipation of energy, are the cause of the evolution of the system's rotational motion. Approximate equations describing this evolution are derived, together with averaged equations in Andoyer variables.

Study of the terms of coupling between rotational and translational motions

The equations of the translational-rotational motion of an Earth's artificial satellite, taking into account only the gravitational attraction, can be put in canonical form using the Delaunay and Andoyer variables. Terms due to the oblateness of the Earth will be considered, as well as, different moments of inertia of the satellite, when it is very close to another. Hori's Method is applied to analyse the equations and the solution is presented up to the first order.

On the rotational motion of a solid carrying a viscoelastic disc in a central field of force

The motion of a mechanical system consisting of a symmetrical solid and a round plate (disc) located in the equatorial plane of the ellipsoid of inertial of the solid is considered. It is assumed that the centre of mass of the system moves in a circular orbit in a Newtonian field of force. The disc flexural deformation, accompanied by the dissipation of energy, are the cause of the development of rotational motion in the system. Approximate equations that define this development are obtained, using the method of motion separation and of averaging /1–3/. The averaged equations that define the evolution in Andoyer variables are similar to the equations that describe the evolution of motions of a satellite with flexible viscoelastic rods located along the axis of symmetry of the satellite /3/.