A method of regularizing the equations of motion in the central force-field

This paper deals with a method of regularization and linearization of the equations of motion in the central force-field, when the potential is given.This method of regularization of the equations of motion is known (Sundman, 1913), and is based on the transformation of time by means of introducing a new independent variable.In this article a condition has been obtained for the regularizing function when the potential is given.Some examples of the perturbed Keplerian motions are discussed.     

An exact linearization and decoupling of the integral equations satisfied by time-dependent X- and Y-functions

We discuss a simple method of linearization and decoupling of the integral equations satisfied by time-dependentX - andY -functions which play an important rôle in the study of non-stationary radiative transfer problems.     

The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

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.     

The Ω dependence in equations of motion

The equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, Ω, and the cosmological constant, Λ. The new equations are expressed in terms of a time variable τ=ln D , where D is the linear rate of growth of density fluctuations. The dependence on the density parameter is proportional to ε=Ω^−0.2−1 times the difference between the peculiar velocity (with respect to τ) of particles and the gravity field (minus the gradient of the potential); or, before shell-crossing, times the sum of the density contrast and the velocity divergence. In a one-dimensional collapse or expansion, the equations are fully independent of Ω and Λ before shell crossing. In the general case, the effect of this weak Ω dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with Λ7ne;0, this enhancement is less pronounced than in an open universe with Λ=0 and the same Ω. Using the spherical collapse model, we find that the increase of the rms density fluctuations in a low-Ω universe relative to that in a flat universe with the same linear normalization is ∼0.01ε(Ω)〈δ³〉, where δ is the density field in the flat universe. The equations predict that the smooth average velocity field scales like Ω^0.6, while the local velocity dispersion (rms value) scales, approximately, like Ω^0.5. High-resolution N -body simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Haloes in an open model simulation are more concentrated than haloes of the same M /Ω in a flat model simulation.     

Numerical integration of the equations of motion of celestial mechanics

Recently a number of new techniques have been developed for the numerical solution of the differential equations governing the motion of bodies in the Solar System, moving under their mutual gravitational forces. Some of these new methods are tested against each other and against more traditional methods and conclusions are made as to under what circumstances any of these methods should be used to produce optimum results.     

A note on the self-consistency of the EIH equations of motion

The self-consistency of the Einstein Infeld and Hoffman (EIH) equations of motion is critically examined in the limiting case of a threebody problem where two bodies are very close to each other and a third quite far removed from them     

Constants of motion of geodesics equations. Examples

Particular examples of constants of motion associated with non-Noetherian symmetries are found for pp-waves, Gödel cosmological solution and Kimura metric. Examples of symmetries for these cases are also obtained.     

Computerized series solution of relativistic equations of motion

A method of solution of the equations of planetary motion is described. It consists of the use of numerical general perturbations in orbital elements and in rectangular coordinates. The solution is expanded in Fourier series in the mean anomaly with the aid of harmonic analysis and computerized series manipulation techniques. A detailed application to the relativistic motion of the planet Mercury is described both for Schwarzschild and isotropic coordinates.Receipt delayed by the postal strike in Great Britain.     

The equations of motion of an artificial satellite in nonsingular variables

The equations of motion of an artificial satellite are given in nonsingular variables. Any term in the geopotential is considered as well as luni-solar perturbations up to an arbitrary power ofr/r, r being the geocentric distance of the disturbing body. Resonances with tesseral harmonics and with the Moon or Sun are also considered. By neglecting the shadow effect, the disturbing function for solar radiation is also developed in nonsingular variables for the long periodic perturbations. Formulas are developed for implementation of the theory in actual computations.     

Two-body motion under the inverse square central force and equivalent geodesic flows

In this paper the fixed energy surfaces for the two-body problem for parabolic and, in particular, hyperbolic motion are completely, determined by utilizing an earlier work of J. Moser. The characterization of these fixed energy manifolds yields the explicit solutions to the above problems in an elementary way for arbitrary dimensions.     

Regularization in the Ideal Resonance Problem

The Ideal Resonance Problem, defined by the HamiltonianF=B(y)+2A (y) sin² x, 1, has been solved in Garfinkelet al. (1971). There the solution has beenregularized by means of a special function _j , introduced into the new HamiltonianF, under the tacit assumption thatA anB¨' are of order unity.This assumption, violated in some applications of the theory, is replaced here by the weaker assumption ofnormality, which admits zeros ofA andB inshallow resonance. It is shown here that these zeros generate singularities, which can be suppressed if _j is suitably redefined.With the modified _j , and with the assumption of normality, the solution is regularized for all values ofB, B¨', andA. As in the previous paper, the solution isglobal, including asymptotically the classical limit withB as a divisor of O(1).A regularized first-order aloorithm is constructed here as an illustration and a check.Presented at the XXII International Congress of I.A.F., Brussels, Belgium, Sept. 20, 1971.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

A note on a separation of the linearized equations of motion in the elliptic restricted problem

Some additional remarks are made here concerning an exact proof of the applicability of Hill's equations in the linear-stability question in the elliptic restricted problem of three bodies.     

On Pfaff's equations of motion in dynamics; Applications to satellite theory

In this article we study a form of equations of motion which is different from Lagrange's and Hamilton's equations: Pfaff's equations of motion. Pfaff's equations of motion were published in 1815 and are remarkably elegant as well as general, but still they are much less well known. Pfaff's equations can also be considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangian which is quadratic in the velocity components. The article first treats the theory of changes of variables in Pfaff's equations and the connections with canonical equations as well as canonical transformations. Then the applications to the perturbed two-body problem are treated in detail. Finally, the Pfaffians are given in Hill variables and Scheifele variables. With these two sets of variables, the use of the true anomaly as independent variable is also considered.     

Eigenvalues and eigenvectors for hybrid coordinate equations of motion for flexible spacecraft

Literal characterizations are developed for the eigenvalues and eigenvectors of a system of linear time-invariant equations which describes the attitude motion of flexible spacecraft in terms of hybrid coordinates. The eigenproblem is shown to reduce to that of a symmetric and positive definite matrix of lower dimension. For the zero damping case, both analytical and minimax characterization methods prove to be useful in localizing the eigenvalues, and eigenvectors for systems of large dimension are obtained explicitly in terms of a 3×1 matrix whose elements are available from a system of three algebraic equations provided.     

An alternate transition from the Lagrangian of a satellite to equations of motion

A procedure is outlined for a conservative system which makes it possible to go from a Lagrangian of a librating system to the corresponding equations of motion in the Eulerian form. The transition does not require a choice of rotational coordinates and makes use of angular velocities and direction cosines directly. The procedure thus synthesizes attractive features of two classical approaches and has far reaching consequences: it is particularly useful in formulating equations of motion for complex flexible systems of contemporary interest. For the case of a satellite with two flexible plate-type appendages, for example, the approach reduced the formulation time to one-third. The basic mathematical concepts are briefly touched upon in the beginning which help explain the subsequent development.