Point-connected rigid bodies in a topological tree

Identical equations of motion are shown to emerge for a system ofn+1 rigid bodies all interconnected byn points, each of which is common to two bodies, by means of each of the following derivation procedures, all of which employ a kinematical identity developed by Hooker and Margulies: The Hooker-Margulies/Hooker equations; Kane's quasicoordinate formulation of D'Alembert's principle; the combination of Lagrange's generalized coordinate equations and Lagrange's quasicoordinate equations; and the combination of Lagrange's generalized coordinate equations and the vector rotational equationM=H applied to the total system and resolved into a vector basis fixed in a reference body of the system. Thus the previously published Hooker-Margulies/Hooker equations are shown to be the natural result of several derivation procedures other than the Newton-Euler method originally used, provided that the central kinematical identity of the original derivation of Hooker and Margulies is employed.     

Matrix perturbation methods using regularized coordinates

Matrix methods for computing perturbations of non-linear perturbed systems, as formulated by Alexeev, involve an expression for the full solution of the first variational equations of the system evaluated about a reference orbit. These cannot be immediately applied to a regularized system of equations where perturbations about Keplerian motion are considered since the solution of the variational equations of regularized Keplerian motion does not in general correspond to the solution of the variational equations of the unregularized equations. But, as Kustaanheimo and Stiefel have pointed out, the regularized equations of Keplerian motion should be excellent for the initiation of a perturbation theory since they are linear in form. This paper describes a method for applying Alexeev's theorem to a regularized system where full advantage is taken of the basic linear form of the unperturbed equations.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Thin-Shell Magnetohydrodynamic Equations for the Solar Tachocline

We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity.     

The nth-order stellar hydrodynamic equation: transfer of comoving moments and pressures

The exact mathematical expression for an arbitrary n th-order stellar hydrodynamic equation is explicitly obtained depending on the central moments of the velocity distribution. In such a form the equations are physically meaningful, since they can be compared with the ordinary hydrodynamic equations of compressible, viscous fluids. The equations are deduced without any particular assumptions about symmetries, steadiness or particular kinematic behaviours, so that they can be used in their complete form, and for any order, in future works with improved observational data. Also, in order to work with a finite number of equations and unknowns, which would provide a dynamic model for the stellar system, the n th-order equation is needed to investigate in a more general way the closure conditions, which may be expressed in terms of velocity distribution statistics. A case example for a Schwarzschild distribution shows how the infinite hierarchy of hydrodynamic equations is reduced to the equations of orders   n = 0, 1, 2, 3  , owing to the recurrent form of the central moments and to the equations of order   n = 2  and 3, which become closure conditions for higher even- and odd-order equations, respectively. The closure example is generalized to a quadratic function in the peculiar velocities, so that the equivalence between moment equations and the system of equations that Chandrasekhar had obtained working from the collisionless Boltzmann equation is borne out.     

Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

On the possibility of studying the dynamics of astrophysical disks in a two-dimensional formulation

A closed system of two-dimensional equations describing the dynamics of rotating, gravitating gas disks is derived. It is an integrodifferential system for barotropic disks and a differential system for polytropic disks. For both barotropic and polytropic disks, these equations differ both from the dynamical equations used in the literature for astrophysical disks and from the traditional equations of two-dimensional hydrodynamics. The sufficient conditions under which the dynamics of a disk can be described in a two-dimensional formulation are obtained. The first condition reflects the thin-disk approximation. The second condition imposes a limit on the characteristic times of processes studied in a two-dimensional formulation. In most cases this condition limits the characteristic frequency of a process to the disk's rotational frequency.Translated from Astrofizika, Vol. 39, No. 3, pp. 441–466, July–September, 1996.     

Hamilton's equations of motion for non-conservative systems

This paper deals with the Hamilton equations of motion and non conservative forces. The paper will show how the Hamilton formalism may be expanded so that the auxiliary equations for any problem may be found in any set of canonical variables, regardless of the nature of the forces involved. Although the expansion does not bring us closer to an analytical solution of the problem, it's simplicity makes it worth noticing.The starting point is a conservative system (for instance a satellite orbiting an oblate planet) with a known Hamiltonian (K) and canonical variables {Q, P}. This system is placed under influence of a non-conservative force (for instance drag-force). The idea is then to use, as far as possible, the same definitions used in the conservative problem, in the process of finding the auxiliary equations for the perturbed system.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

Lagrange variational equations from Hori's method for canonical systems

Hori, in his method for canonical systems, introduces a parameter through an auxiliary system of differential equations. The solutions of this system depend on the parameter and constants of integration. In this paper, Lagrange variational equations for the study of the time dependence of this parameter and of these constants are derived. These variational equations determine how the solutions of the auxiliary system will vary when higher order perturbations are considered. A set of Jacobi's canonical variables may be associated to the constants and parameter of the auxiliary system that reduces Lagrange variational equations to a canonical form.     

A theory of integration for the extended Delaunay method

In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincaré. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincaré. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincaré is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincaré.     

Relaxation Oscillations in Tidally Evolving Satellites

We study the rotational evolution under tidal torques of axisymmetric natural satellites in inclined, precessing orbits. In the spin- and orbit-averaged equations of motion, we find that a global limit cycle exists for parameter values near the stability limit of Cassini state . The limit cycle involves an alternation between states of near-synchronous spin at low obliquity, and strongly subsynchronous spin at an obliquity near 90°. This dynamical feature is characterized as a relaxation oscillation, arising as the system slowly traverses two saddle-node bifurcations in a reduced system. This slow timescale is controlled by ε, the nondimensional tidal dissipation rate. Unfortunately, a straightforward expansion of the governing equations for small ε is shown to be insufficient for understanding the underlying structure of the system. Rather, the dynamical equations of motion possess a singular term, multiplied by ε, which vanishes in the unperturbed system. We thus provide a demonstration that a dissipatively perturbed conservative system can behave qualitatively differently from the unperturbed system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Error estimates with L1 solutions

A method is given, based on the pseudoinverse of the equations of condition, to obtain error estimates for the solution in the normL ₁ of an over-determined linear system. The computational labor to obtain the errors, while not trivial, is less than that for various competing methods, particularly if there are many more equations of condition than unknowns. The error estimates for anL ₁ solution are substantially larger than those for a least squares solution of the some data. It is suggested that a complete discussion of a linear system include at least bothL ₁ and least squares solutions with their respective errors and the condition number of the linear system.     

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.     

Bianchi types of Lie point symmetries arising for differentially rotating stationary axisymmetric fluids

The Einstein field equations for a perfect fluid with two commuting Killing vectors, which span the fluid's four-velocity, are considered. A third space time symmetry, which is a homothetic or a Killing vector, can be used to reduce these equations to a system of ordinary differential equations. This symmetry restricts the form of the differential rotation Ω of the fluid. A Bianchi classification of the resulting Lie algebras is performed and related to the kinematical properties of the fluid.     

Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

Averaging on the motion of a fast revolving body. Application to the stability study of a planetary system

Exploring the global dynamics of a planetary system involves computing integrations for an entire subset of its parameter space. This becomes time-consuming in presence of a planet close to the central star, and in practice this planet will be very often omitted. We derive for this problem an averaged Hamiltonian and the associated equations of motion that allow us to include the average interaction of the fast planet. We demonstrate the application of these equations in the case of the μ Arae system where the ratio of the two fastest periods exceeds 30. In this case, the effect of the inner planet is limited because the planet’s mass is one order of magnitude below the other planetary masses. When the inner planet is massive, considering its averaged interaction with the rest of the system becomes even more crucial.     

About Tidal Evolution of Quasi-Periodic Orbits of Satellites

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. The modern ansatz in the theory of tidal dissipation in Saturn was developed previously by the international team of scientists from various countries in the field of celestial mechanics. Our applying to the theory of tidal dissipation concerns the investigating of the system of ODE-equations (ordinary differential equations) that govern the orbital evolution of the satellites; such an extremely non-linear system of 2 ordinary differential equations describes the mutual internal dynamics for the eccentricity of the orbit along with involving the semi-major axis of the proper satellite into such a monstrous equations. In our derivation, we have presented the elegant analytical solutions to the system above; so, the motivation of our ansatz is to transform the previously presented system of equations to the convenient form, in which the minimum of numerical calculations are required to obtain the final solutions. Preferably, it should be the analytical solutions; we have presented the solution as a set of quasi-periodic cycles via re-inversing of the proper ultra-elliptical integral. It means a quasi-periodic character of the evolution of the eccentricity, of the semi-major axis for the satellite orbit as well as of the quasi-periodic character of the tidal dissipation in the Planet.