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II. Classical Dynamics of the Planetary Motion

As long as the Classical Mechanics is involved – and we shall limit ourselves only to this theoretical environment – the planetary motion is described as a Kepler motion by the vectorial ordinary differential equation

(2.01)

Here K is a constant, denotes the position vector of the material point whose motion is considered with respect to the center of force, and a dot over symbol means derivative upon time. The constant K does not depend on quantities related to the point in motion but only in the case when electric forces are involved. It is easily seen that this motion takes place in a plane. Indeed, by direct vector multiplication of (2.01), we have

(2.02)

showing, in another form, that the acceleration vector is central. On the other hand, we have the identity

(2.03)

This identity shows, first of all that the normal to the plane supported by the position and velocity vectors is preserved along the motion, i.e. the motion is plane. Secondly, equation (2.02) actually expresses the conservation of the vectorial rate of area swept by the position vector. In other words, the plane of motion as well as the magnitude of the area rate are conserved all along the trajectory. In view of this we can simplify the geometry by confining it to the plane of motion, where the coordinates of the material point in motion are ξ and η say (Lawrence, Mittag, 1992). Equation (2.01) is then equivalent to the system

(2.04)

with r and f the polar coordinates of the plane with respect to the attraction center as origin. The magnitude of the rate of area swept by the position vector of the particle is then given by

(2.05)

This constant of motion allows us an elegant integration of the system (2.04) with the analytical form of the trajectory as a direct outcome. First we define the complex variable

(2.06)

so that (2.04) can be written as

(2.07)

Now use equation (2.05) to eliminate r² with the result

(2.08)

where m º m₁ + im₂ is a complex constant to be determined by the initial conditions of the problem. The analytical equation of motion can be then directly extracted from (2.05) by using (2.08). In polar coordinates of the plane of motion the result is

(2.09)

The shape of this trajectory is best pictured by going back to Cartesian coordinates, where we have, instead of (2.09) the second - degree curve – a conic:

(2.10)

The center of this conic is not the center of the force, but has the coordinates

(2.11)

In cases where Δ = 0, the center of this trajectory is at infinity: the trajectory is a parabola. We have here the ballistic cases, where the basic motion is parabolic. Of course, in reality the center of such a trajectory is not exactly at infinity but at a distance much larger than the maximum height of the trajectory. Thus the real ballistic motion is only an approximation of the ideal case, the smaller maximum height the better the approximation.

      Assuming the center of the trajectory at finite distance, and referring the trajectory to this center by the translation x = ξ – ξ₀, y = η – η₀, its equation becomes

(2.12)

The quadratic form from the left hand side of this equation is completely characterized by the 2×2 matrix

(2.13)

The eigenvalues of this matrix are Δ and () with the corresponding eigenvectors

(2.14)

Thus, the orientation of trajectory in its plane is completely defined by the initial conditions of the motion. The magnitude ‘m’ is proportional with the eccentricity ‘e’ of trajectory. Indeed the semi axes ‘a’ and ‘b’ are

(2.15)

Thus, the initial conditions can actually be expressed only in terms of ‘contemporary’ magnitudes allowing us to forget about the past:

(2.16)

One last remark of classical provenance: as it can be seen directly from equations (2.15) one of the semi axes can be imaginary, for Δ < 0, in which case we have to do with hyperbolic trajectories. It is only in cases where Δ > 0, that we have to do with elliptic trajectories, properly representing planetary motion. Thus, the parabolic trajectories are all characterized by points on the circle Δ = 0, i.e.

(2.17)

and the whole interior of this circle corresponds to all possible finite motions that a material point can have around a center of force acting with a force inversely proportional to square of distance. This would mean that a planet for instance would have infinitely many possible initial conditions we have to choose from. This is a freedom hard to take, for the usual wisdom maintains that the actual motion of a planet is perceived as if it had unique initial conditions. It is the departure from this perception which has induced arguments about all kinds of actual perturbations acting on the planet. To a certain extent this is true: the discovery of Neptune is an example. However it is not universal, and our objective here is to dig in the apriori basis of this freedom. It is much like the apriori possibility of Geometries, with which our subject has everything in common.

     Indeed, even superficially it can be seen at once that the mentioned freedom of choice of initial conditions allows us to construct a Cayleyan (Absolute) Geometry based on the possibility of variation of the trajectories. We know that an Absolute Geometry is related to some conservation laws, at least as long as some realizations of SL (2,R) group structure are involved. And indeed, the absolute metric for the interior of the circle (2.17)

(2.18)

can be brought to the form of Poincaré metric

(2.19)

by the following transformation of coordinates:

(2.20)

It is thus apparent that we can use all the advantages related to this Geometry in order to draw conclusions on families of Kepler motions. Different families can correspond to different physical structures. Some examples are discussed elsewhere. What we need now is to see what kinds of families of motions are controlled by a gravitational field at a grand scale.

     In order to illustrate the general intention here, notice that the classical dynamical problem of planetary motion is considered solved once we have proper initial conditions for the motion of a planet. This is actually a mathematical characteristic of every physical problem whose solution depends on an ordinary differential equation. Fact is that if we are to read natural characteristics in the solutions of differential equations we may be greatly misled, as is actually the case with the Newtonian description of a planet’s motion. First and foremost we got nowadays an astronomical perspective of the problem, and from this point of view the motion of planets is far from being planar. However, as long as we consider the gravitational force as a central force, this planeness of the motion is theoretically inescapable. The characteristic of centrality of the gravitational force seems nevertheless indicated by our daily experience, and its extrapolation to the planetary scale seems only rational. A way to make the Newtonian model cope with astronomical perspective is either by inserting considerations from Special Relativity regarding the variation of mass or by cosmological considerations regarding the variation of fundamental constants, and these paths of research have already been considered in many variants within the very Classical Mechanics. There is still another way, quite radical we might say, that of General Relativity: the motion is a space-time property of matter. Then the Kepler motion appears as a natural generalization of the free motion in the Newtonian acceptance. Many attempts along general relativistic ideas have also been historically made and, again, none has ever succeeded completely. Worse than this, every attempt along these lines left in its wake a host of problems that seem to multiply by the day. What we think is to be positively taken into consideration from all the historical attempts to update the Newtonian theory regarding the Kepler motion, is that they indicate the necessity of an opening of the model in order to accommodate our natural tendency to describe the Universe hierarchically: a planetary system is itself a synthesis, while being the fundamental brick of a bigger system.

     Now, it is a matter of particular opinion where the opening is placed into the model, and this fact is the imprint for each classical improvements of the problem. We hold it true that the opening should be placed exactly where the freedom is, i.e. in the initial conditions of the Kepler motion. And not only for the ‘mathematical beauty’ as reflected in the group structure above but also for physical reasons of putting an old idea on a formal basis: the parabolic motion, which is a fact of daily experience and was used by Newton as a model in building the laws of dynamics, is also a reference geometrically. Indeed, when a Kepler motion is represented in terms of its initial conditions, it is represented by a point belonging to a 2D manifold. In this manifold the parabolic motions are represented by points situated on the Cayleyan Absolute. One can say that the Newtonian program of concluding from parabolic motion to planetary motion is paralleled by a geometrical process of describing the kinematics of the points internal to the Absolute, based on the motion of the points of Absolute. This geometrical image will be taken into consideration elsewhere. In the remainder of this essay we will try to use it only in order to couple the Kepler motion with inertia.

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