Kepler motion

If we consider the sun and one planet, namely a two-body case, the equations of motion for this case is solvable. We have the famous Kepler motion.lt is well known that there are four types of orbits, namely the circle (e = 0), the elliptic curve (0 < e < 1), the parabolic curve (e = 1), and the hyperbolic curve (e > 1), where e is the eccentricity. 

Kustaanheimo, P. and Stiefel, E. (1965) Perturbation theory of Kepler motion based on spinor regularization, J. Reine Angew. Math 218, 204-219... 

Abstract In order to describe the motion of two weakly interacting satellites of a central body we suggest to use orbital elements based on the the linear theory of Kepler motion in Levi-Civita s regularizing coordinates. The basic model is the planar three-body problem with two small masses, a model in which both regular (e.g. quasi-periodic) as well as chaotic motion can occur. 

This paper discusses the basics of this approach and illustrates it with a typical example. First, we will revisit Levi-Civita s regularization of the two-dimensional Kepler motion and introduce sets of orbital elements based on the differential equations of the harmonic oscillator. Then, the corresponding theory for the three-dimensional motion will be developed using a quaternion representation of Kustaanheimo-Stiefel (KS) regularization we present it by means of an elegant new notation. 

We begin by summarizing the equations of motion of the three-body problem with two small masses in the form of two weakly coupled Kepler motions, valid in two or three dimensions. 

Levi-Civita Regularization of Perturbed Kepler motion... 

In order to apply one of the proposed sets of elements for describing coorbital motion we formulate the equations of motion (1) of the weakly coupled Kepler motions in terms of the Levi-Civita coordinates of Section 2. In this way the unperturbed problem will be defined by linear differential equations. Using the symbols pj = mo + nij, j = 1,2 as well as complex notation iq, r2 C and the abbreviations fi, f2 C for the right-hand sides, equation (1) reads as... 

In order to regularize the perturbed three-dimensional Kepler motion by means of the KS transformation it is necessary to look at the properties of the map (42) under differentiation. 

In the preceding text we have presented a unified theory of regularization of the perturbed Kepler motion. Quaternion algebra allows for an elegant treatment of the spatial case in a way completely analogous to the way the planar case is traditionally handled by means of complex numbers. As a consequence of the linearity of the regularized equations of the perturbed Kepler motion, the problem of satellite encounters reduces to a linear perturbation problem, the problem of coupled harmonic oscillators. Orbital elements based on the oscillators may lead to a simpified discussion of ordered and chaotic behavior in repeated satellite encounters. This has been demonstrated by means of an instructive example. 

Paul Kustaanheimo (1964). Spinor regularization of the Kepler motion. Ann. Univ. Turku, Ser. AI 73. 

In the ease of the Kepler motions the Fourier series for the rectangular co-ordinates , rj and for the distance r are comparatively easy to find. Noting that rja and ja are even functions, and rjja an uneven function of u, and therefore also of wlt we can put... 

The equation has almost the same form as in the case of the Kepler motion A and B have the same meaning as there ... 

This equation differs from the corresponding one in the nou-relativ-istic Kepler motion by the term... 

H0 is the Hamiltonian function of the non-relativistic Kepler motion, which we regard as the unperturbed motion, and Hx is a perturbation function. In order to find the influence of this perturbation on the Kepler motion, we have to average Hx over the unperturbed motion. If we express the sum of the squares of the momenta occurring in Hx with the help of the equation for W0, we obtain... 

For E =0 the motion of the Stark effect passes over into the simple Kepler motion. This is separable in polar co-ordinates as well as in parabolic co-ordinates. From the separation in polar co-ordinates ( 22) we obtain the action variables Jr, Je, J0, and the quantum condition... 

If now we calculate the Kepler motion in parabolic co-ordinates, we have only to put E=0 in the above calculations. We obtain the action variables J(, J, and (the last has the same significance as in polar co-ordinates) and the quantum condition... 

In place of the single quantum state characterised by a single n, as in the case of the Kepler motion in the absence of a field, we have the 2n—1 states already mentioned in 35. 

As already mentioned, the accidental degeneration of the unperturbed system is a very exceptional case in astronomy. In atomic physics, on the other hand, it plays an important role, for firstly, according to Bohr s ideas, a whole set of equivalent orbits occur in the higher atoms and again according to the quantum theory the periods of rotation of the Kepler motions with different principal quantum numbers are always commensurable, since they vary as the cubes of whole numbers. 

An example will at once make clear what is meant. Let us take the relativistic Kepler motion, or, in other words, motion in an ellipse with a perihelion rotation. In general, the path fills a circular ring and, therefore, a two-dimensional region, densely everywhere. The boundaries for the libration of the radius vector are here concentric circles. 

The expansions of the cartesian co-ordinates as functions of the angle variables (to be calculated from (26), 22) must now be introduced, to provide a starting-point for the calculation of the perturbations. In this connection, however, there is one point to be borne in mind. In the unperturbed Kepler motion (without taking account of the variation in mass) only Jx is fixed by the quantum theory, whilst J2, i.e. the eccentricity, remains arbitrary in the relativistic Kepler motion, J2 is also to be quantised and, for a one-quantum orbit, J2=J1=A. We shall not take account quantitatively of the relativistic variation of mass, but we shall assume that the initial orbit of each electron is circular with limiting degeneration J1=A,... 

Let us resolve this function into H0 and H1, where H0 is the Hamiltonian function of the (non-relativistic) Kepler motion of the inner electron and Hx the remaining part of the above expression. 

We must now introduce into Hi the angle and action variables wly w2, Jlt J2 of the unperturbed Kepler motion of the outer electron, represented by the term U0. We shall, however, replace wl by the true anomaly 1 which is connected with w1 by the equation... 

Atomic Structure and Chemical Properties 31. The Actual Quantum Numbers op the Optical Terms 32. The Building Up op the Periodic System op the Elements. 33. The Relativistic Kepler Motion. ... 

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