KINEMATICS OF FLUXES AND BUNDLES OF TRAJECTORIES
Author: Nicolae Mazilu
Published on Thursday, April 29th, 2010 in category ProtoQuant
INTRODUCTION
The revolution of a planet around the Sun can be characterized by a toroidal bundle of orbits of its component material points. The flux of this bundle through any cross-section of the bundle is continuously changing, and we can assign this variation to quite a few geometrical phenomena. First of all, assuming that the bundle has a constant flux of orbits, a material point occupying a certain orbit in a cross-section of the bundle never occupies the same orbit in the following cross-section, due to the processes of rotation and deformation of the planet. Secondly, the flux itself may not be conserved along the toroid, so that a certain material point from the physical structure of the planet, occupying a certain orbit in a cross-section of the toroid, may occupy an entirely new orbit in the immediate following cross-section: this way an orbit is born, enriching the flux. On the other hand, one can hardly assume that the physical structure of the planet is preserved in detail, so that it appears quite naturally that new material points are born during the revolution of the planet around the Sun. Thus we can say that there are basically two causes of geometric variation of the tube representing a planet in motion around the Sun: one due to the variation of the flux of orbits representing the trajectories of the structural material points, the other due to the variation of the physical structure of the planet itself.
Likewise, a magnetic flux tube, of force free magnetic field lines is completely analogous to the situation of a planet around the Sun. Only here the lines themselves receive a kind of more “objective” character, justifying the idea of a “kinematics” of them, in order to describe phenomena such as knotting of lines and the formation of flux ropes, considered today to be responsible for the solar unrest. Here too, we need to discern between the description of the flux of lines itself, and that of the flux of particles following those lines. It is felt here that part of the evolution of the flux rope is due to the interaction between the particles and field, and that the lines of field are an object of dynamics exactly like the particles themselves.
In order to be able to write a dynamics of these complex physical situations, we have first to write a kinematics, in order to know how and by what channals the dynamics should be rationalized. And because the magnetic solar formations can be supposed of the same dynamical origin as the planetary motion, an exercise will be made here to describe these phenomena by the same mathematical formalism. The orbits of material points in the case of planet and the lines of field will be described by the same equation - an ellipse referred to its center. In this case each “line” is a geometrical object characterized by three coordinates, and “particle” is described by two coordinates. Thus a generic line is given by the constraint:
 |
(1) |
The variation of this line can be described as a process of plane deformation, which adds something in the coefficients. So, let us say that a new quadratic form is expressed by
 |
(2) |
where λ and μ are two parameters reflecting some external physical conditions, and δ denotes a pure variation of the parameters to which it is applied. It is therefore important to see how we can express the variation of quadratic form as due solely to the variation of the coefficients.
THE DEFORMATION OF LINES
We borrow now heavily from the theory of deformation of the surfaces. The coordinate deformation is defined in the following way: let’s assume that the initial orbit undergoes a small deformation, so that its vector is given by
 |
(3) |
with ε a small quantity. Then the deformation is infinitesimal, if we have satisfied the condition
 |
(4) |
where we denoted with α the matrix of coefficients of (1), and used the Dirac notation. This is indeed equivalent to vectorial notation but helps to find a little easier the general relations. The condition (4) imposes some definite restrictions on the vector 
. To see what these restrictions are, let us write the quadratic form corresponding to the vector from equation (3). We have
 |
(5) |
and equation (4) shows that this deformation is infinitesimal if, and only if, the condition 
is satisfied, i.e. if the two vectors are “orthogonal”. This means that the vector of deformation is defined up to a normalization factor: if ξ, η are its components, we have
 |
(6) |
Therefore, the infinitesimal displacement generated by this infinitesimal deformation must proceed along the normal to the line in its plane. The vector product of the position with the vector of deformation is oriented normally to the plane of motion, and has the magnitude given by the very quadratic form of the conic:
 |
(7) |
The vector from equation (6) has parameter participation in its structure. More to the point, if one takes the vector tangent to orbit given by
 |
(8) |
and builds the vector product, one obtains
 |
(9) |
There should be therefore always a force, due to the variation of the parameters, which drags the line itself perpendicular to its plane.
A SPECIAL EVOLUTION
The problem we need to solve is the following: in what conditions an infinitesimal displacement (dx, dy) of a particle located on a given orbit results in a point located on another orbit of the same center? The condition can be written in the primary form as
 |
(10) |
This expresses the fact that the point characterized by the position (x, y) on the first ellipse, goes into the point (x + dx, y + dy) which is located on the second ellipse, having the parameters slightly different from the first. Expanding the second relation (10) and taking into consideration the first one, we end up with the equation
 |
(11) |
On the other hand, from the equation of the first ellipse we have
 |
(12) |
on the assumption that the displacement coincides with the differential. The equation (11) then simplifies to
 |
(13) |
It is quite obvious that in equations (12) and (13) there are two pairs of identical terms, in view of the symmetry of the matrix α. We keep however the original form in order to discern between them in case the infinitesimal displacement is given by an equation of evolution. Assuming, indeed, that there is an evolution given by a matrix, Ω say, this matrix doesn’t necessarily need to be symmetric. The evolution is defined by equation
 |
(14) |
The equation (13) can then be referred to the same arbitrary generic position, because we have
 |
(15) |
The problem is now to find the matrix of evolution Ω, satisfying the matrix equation
 |
(16) |
as a function of the matrix of parameters and their differentials. The last monomial represents a third order differential, while all the others are second order differentials. The equation can then be simplified to a homogeneous equation of degree 2 by neglecting the third order term. We therefore have
 |
(17) |
The first of these equations is the equivalent to the equation (12) where the evolution (14) is applied. Now, using the first of equations (17) to simplify the second one, we get for the matrix of evolution Ω the following equation
 |
(18) |
This matrix equation is identically satisfied by the following matrix of evolution
 |
(19) |
which expresses the fact that evolution (14) represents the equivalence between variation of the parameters in the same point, and the displacement in the neighborhood. In detail equation (19) gives
 |
(20) |
where e is the identity 2×2 matrix and ω1,2,3
are the following differential forms
 |
(21) |
First of all some differential properties of these 1-forms. If we differentiate exterior them, we get the following equations:
 |
(22) |
where Ω is the following differential 2-form:
 |
(23) |
This 2-form has the property of being an exact differential:
 |
(24) |
This can be easily verified by direct calculation. Indeed it can also be easily verified by direct calculation that Ω is the exterior derivative of the 1-form
 |
(25) |
which proves directly the assertion. The 1-form Ψ is the Hannay angle of the tube of lines. Here we have a SL (2, R) group algebra characterized by the relations of structure
 |
(26) |
Now we need some space properties related to these differentials. First note that the quadratic form
 |
(27) |
is conjugated with respect to the quadratic form from equation (1). This means that the two families of orbits are related by a certain property of orthogonality. But more importantly the vector
 |
(28) |
from the plane of original orbit has the following characterization with respect to the position vector:
 |
(29) |
Consequently, the displacement in matter, as characterized by vector (27), makes an angle with the position vector whose principal value, θ say, is given by equation
 |
(30) |
The factors of this ratio have indeed the essential property of sine and cosine.
THE FLUX OF LINES PROPER
The 2-form Ω from equation (22) is a flux in the space of the parameters of the orbits. If we choose to represent the orbits in the form given in equation (1) then the parameters of the trajectory are non-homogeneous coordinates representing an entire orbit. Then the problem of rational representation of an extended body, or a tube of magnetic lines, depends on the value of the integral from the form Ω and, more to the point on its variation. This variation is given by the theorem of Betounes
 |
(31) |
This time the vector 
denotes the ensemble of parameters (α, β, γ), and the function Φt is an evolution of this ensemble. The domain D is the interior of the toroid and the surface S is its surface. The speed 
is simply the speed expressed in these line coordinates. In view of the equation (23) the second term in equation (30) is null, so that the formula becomes
 |
(33) |
The problem now is to calculate the dot product in the right-hand side of this equation. It is dictated by the “vector product” of the velocity with the “position”:
 |
(34) |
The problem is to express this product on the toroid. The value of this 1-form depends on the components of the vector . It is always zero if in cases where is proportional with (α, β, γ) or with their differentials. Consequently, nontrivial results must occur with nontrivial velocity definitions. In the case of solar perturbations, these nontrivial definitions appear to be related with the planets of the solar system, which generate periodic evolutions of the solar activity.