Absract

As was demonstrated in "Riemann configuration spaces", the form of the distance element Lagrangian in the multidimensional configuration space of observables which recovered the classical Lagrangian, T-V, when the realm of everyday velocities was small compared to c is

L= (1)

The purpose of this investigation will be to determine the equations of motion generated by (1) when the number of observables tends to infinity or the configuration space becomes infinite dimensional. In this scenario the configuration space can be conceived as being a continuum or a fluid.  

Fluid Path

With a discrete number of observables the path of a system as it traces out the shortest distance in the configuration space can be described by x(i,t), which is basically the value of observable at time t and where i is an integer. To elaborate further the path can be thought of as a mapping

X[t]:N-> ,   where N={1,2,3 ....}

ie a mapping which changes continually with t. This concept can be extended to the case where i is itself continuous,  ie  

X[t]:-> .  where S=[0 1]

By allowing the observable label to become continuous we can begin to describe the evolution of a continuum of observations. In the example to follow the system will be a physical fluid confined to move in one dimension. Throughout the generalization to three dimensions will be apparent.

One dimensional fluid

Let

X[λ,t]  , λ ε S

describe the path of the fluid. So the velocity of a particle identified by the label λ is simply



In what follows the total potential energy, φ, and total mass, M, will be treated interchangeably through φ=M so that the mass of a particle can be thought of as simply its contribution to the total potential energy. In this sense since potential energy is conceivably  a function of the particles proximity to one another and hence is a geometrical notion, then so to is mass, and the concept of substance is not necessary. For example imagine a spring composed of a material which in its uncompressed state had virtually no mass and hence no substance. However energy/mass could be added to the spring by simply compressing it, and compression is a well defined geometric notion. It is for the same reason that the constituents of an atom’s nucleus are much more massive than the outer orbiting electrons, by virtue of the fact that they are squeezed much more tightly together to overcome the coulomb repulsion and hence have more potential energy. Objects which are in isolation from the rest of the system tend to contribute less to the overall potential  energy  of the system and hence have less mass which only increases when their proximity to the rest of the system increases. So along the same line of reasoning an object in deep space in isolation to everything else will have less mass. All in all we are headed towards a relational theory of the universe in keeping with Machs principle.
   In the discrete case the total potential, φ, is    

φ[,t]=    (2)

=  j~=i  (3)

Where is the individual contribution from a particle at and is the interaction potential between particles i and j.  We can identify the mass of particle i to be simply

=

as n->∞ we make the assumption that φ  remains bounded or that

(n )  -> as n->∞

which => from (2) that

φ=  .

is the contribution from those particles identified by the range [λ   λ+dλ].

The kinetic energy T can be similarly defined as,

   where u[λ,t] =is the velocity of the partical labled by  λ.

As always we choose the path X[λ,t] which makes the action A=, a  minimum , ie

0=δA= ==

which is equivalent to applying the Euler-Lagrange equations to the Lagrangain density

l =      (4)

in the (λ,t) space in which and   are treated as functions of time and which are independent of the path, X[λ,t] .

(4) can be simplified to read  

l=( (φ-T) - φ ) = ( (φ-T) - φ ) where  the total constant energy , E=. (5)

If the fractional quatities and   are defined then (5) becomes

  l=E (1--)  (6)
  
  Since E is a constant then it can be dropped from (6) without loss of generality to give
  
    l= (1--)                                                       (7)
    
The fractional   quantity is what is measured in an experiment, not the absolute quantity. If the absolute energy in the universe where to uniformly double, no experiment could be devised to detect it, and therefore in any physical theory, only ratios and relational quantities should appear.  Subjecting (7) to the Euler-Lagrange equations gives

                          (8)

Galilean Invariance

Equation (9) is not invariant with respect to  Galilean transformations. Invaraince can be introduced without modifying the equations of motion by instead considering the following distance element Lagrangian and repeating the above analysis.

=                                    (9)

where  P=.  

Under this modification (8) becomes

   (10)

with  =  ,the total kinectic energy , and

= the total momentum.

If we define the force term to be

F=-,

then it is implied from equation (10) that,

                                                  (11)

Also (10) and (11)  are  consistent with the assumption that is a constant quantity, ie assuming that is a constant does
not produce any inconsistencies in (10). In light of this we can always move to a frame of reference in which =0 and therefore recover equation (8). Classically this is a frame in which the center of mass of the system has zero velocity.

From here on in we will define = m[x[λ,t],t],  Ψ[x[λ,t],t]=  and , so that (10) becomes

       (12)
  
     (13)

If the above analysis is repeated for a fluid moving in all three dimensions of the physical background space, and in which the total momentum along each of the three Cartesian axes is zero, for which incidentally a frame of reference can always be chosen , (12)  becomes

   (14)

  with T= and being the vector dot product.
  

Modeling finite particles

Treating the fluid path as a step function allows for the description of what would be classically termed 'solid point particles', e.g.
choosing

X[λ,0]= x1 , 0<λ<

X[λ,0]= x1 , <λ<

will describe the distribution of mass/energy for two classical  point particles of equal mass at time t=0. in Newtonian physics the
distribution of mass/energy would remain the same as the system evolves, i.e. the particles would remain intact. However this cannot be guaranteed by equation (12), and in general equation (12) allows for a spreading of the mass/energy distribution as the system evolves

Similarity to other force laws

Equation (12) is similar to Weber's force law which describes the mutual force between a pair of charges as a function of their
separation and relative velocity and acceleration  

F=

where in this case Ψ is the coulomb potential between the charges and X is there separation   In general   will not be the
same as in three spatial dimensions and (14) does not contain the induction term, . However velocity induction
is possible in (14) even in the absence of  the force term   since

=

In other words, a particle in motion which experience's no force due to can nevertheless still be accelerated through
A variation in its fractional mass m. A change in m could be achieved by changing the energy density in a different part of the
system  since remembering that m is the fractional contribution of the particle to the overall energy of the system. Since
  and    is the typical order of magnitude observed in an experiment, then would be  expected to be very small and hence would not manifest itself in everyday observations. The induction term could allow for the design of
novel propulsion systems in the future. As a simple example, consider a wound up spring, the mass of which is entirely due to the potential in its winding. If after been wound the spring is initially set in motion at a uniform speed, and then at a subsequent point allowed to unwind, the spring will undergo an acceleration.   

Determination of  

from the above analysis it has been assumed that the form of is already know.   essentially  represents the metric coefficients in the infinite dimensional Riemann manifold from which the geodesics are calculated, and which are basically represented by equation (14).  The question becomes what are the conditions under which the form of   is generated.

If one considers the  4-D space-time metric formed by

   (15)

and then follows through on  the vacuum version of the  Einstein-Hilbert approach by  minimizing the action,

S=,
  
where R is the  curvature scalar corresponding to (15) , a wave equation is obtained for    in which

                                       (16)

However the problem with (16) is that it does not contain a source term and is not invariant with respect to Galilean transformations. Attempts to include a source term following the Einstein Hilbert approach have meet with unsatisfactory results even when
the so called 'matter component' is included to form

  S=  
  
  where l is the lagrangian density defined in equation (7)
  
  As an alternative approach but still employing the concept of  least curvature, the curvature of the entire configuration space
  may be defined and then  subject to variation for which the curvature is a minimum. Investigations employing this approach are   currently under way and the results will be published in a subsequent paper.