Dynamics deals with the movement of objects in space.  Therefore it is of fundamental importance to begin with a definition of a relation coordinate system.

The entity relation structure provides the basic formulation for the relation coordinate system.  To define the coordinate system I begin with an explicit measure on the entity relation structure.  The following is based on Watanabe’s [1969] Interdependence Analysis (IDA).

Let  stand for the set of states of entity a and  stand for the set of states of the remainder of the system, thus is the set of states of the entire system.  To each member  of  is assigned a probability.

Let  be the information entropy function on, then

                                                                     (3.1)

Following Watanabe [1969, 52-56], I define

                                                                    (3.2)

as the entropy measure of the interdependence structure between  and . 

If there are n entities then we have

                           (3.3)

The interdependence between any two entities can be written as

                (3.4)

I propose an (n-1) interdependence configuration space for each entity, where each coordinate represents the inverse of the interdependence with one of the other (n-1) entities.  This gives a total set of n(n-1) coordinates.  However, since, the number of independent coordinates is n(n-1)/2. 

Watanabe assumes that the state space of an entity is one dimensional.  If entities exhibit multi-dimensional behaviors then we can simply and an additional index parameter.   is the interdependence between  and  for behavior .

The choice of this configuration space in which relation work, force, energy, momentum, etc.  are defined is of fundamental significance.   Within this configuration space we define such concepts as organization energy, work, force, momentum, and derive Lagrangian and Hamiltonian equations of motions.  Before proceeding with such definitions it is worthwhile to pause and consider the structure of this relation space.  Even though we will be able to define these concepts that mathematically resemble their physical counter parts, transferring our intuitive notion of them from physical reality to relations is often questionable.  Transferring our intuitive understanding of the physical space to relation space is completely inappropriate. 

Regardless of the structure of space and time that may be proposed within more advanced theories, they must reduce to the classical and intuitively understood Euclidean geometry of everyday experience.  There are at least four differences with relation configuration space that makes any attempt of intuitive or graphical understanding in terms of Euclidean geometry misleading.

1. The measure of the relation and therefore the distance is a statistical measure based on many observations about the relation of the entities state space with that of all other entities.  We can think of the measurement as evaluating the statistical properties of a message.  Even if one accept the proposition that the relation has an immediate and real value and the extended observations are only necessary to get an accurate distance measure, we are left with the problem of how does one actually observe changes in relations that are of shorter time duration than can be measured by changes in the statistics of the messages representing the relation.
2. Each entity sees only a subset of the full relations space.  The definition of that subset places that entity in a special entity-centric position.
3. The structure of the relations space in not even Riemannian much less Euclidean.  Consider that an entity A may have a relation with entity B, and B with C.  Thus there is a finite relation distance between A and B; and between B and C.  There may be no relation between A and C, in which case J = 0 and the distance between A and C goes to infinity.  The structure of the space seems to make sense only in the individual subset perspective of each entity.
4. The entire structure of the relation space is dependent upon what we choose to measure.  The IDA measure depends on what parameters of the entities we choose to measure.  Change the parameters and the entire structure may change.  Entities that were close based on one set of parameters may be distant on another set of parameters.

With such major differences is there any value in the concept of relation dynamics? I obviously think so, or I would not be writing this.  However, we will find that the usefulness of applying the concepts of relation dynamics is limited.  The real value will occur when we take the next step to relation thermodynamics.  A discussion of why this occurs and the significant of the value will have to wait until we get there.

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(C) 2005-2014 Wayne M. Angel.  All rights reserved.