11. Generalized Exchange and Laws of Conservation

L. G. Kreidik (translation from Russian T. S. Kortneva and G. P. Shpenkov)

We will consider kinematic exchange between a system and the environment

on the *Z*-level of rest-motion (Fig. 2.15).

**Fig. **2.15. A graph of *Z-level *exchange.

Let motion-rest *
*be transferred from the environment to the system and the amount *
*of motion-rest be transferred from the system to the environment along the
kinetic channel and *
*be transferred by the system over the potential. If ,
then

, , , (2.262)

where *
is a *parameter of any level of motion, *r *is kinetic resistance or
kinetic elasticity, *k *is potential resistance or potential elasticity,
and
are differentials of particular states.

In a general case, the resistances of the exchange channels depend on the state of the system, environment, and the character of the exchange channels; in the linear approximation they are constant. Their inverse values, *g *and *C , *will be called kinetic and potential conductivities, respectively.

Each of the differentials of exchange over a direct and two inverse channels determines the amount of mutual exchange equal to the difference of partial components of exchange. The rest-motion gained by the system is equal to the sum of exchanges in the three channels. Thus, we have

. (2.263)

Hence, we arrive at the equation of exchange in the form:

(2.264)

or

(2.264a)

or

. (2.264b)

The equation of exchange is simultaneously the equation of the state of the system.

We will write the exchange-state equations for -,
-, -
and -* *levels:

, , (2.265)

, (2.266)

or

, , (2.267)

, . (2.268)

In a broad sense, the first terms in the left-hand sides of the equations are kinetic momenta, the second and third terms are kinetic and potential momenta of the feedback with the environment.

If we introduce the generalized charge

, , , (2.269)

where *a *is the characteristic length, then in terms of charges the equation
for the * *-level
becomes:

. (2.270)

For the *
*-level it will be represented by the equation of current:

. (2.271)

Finally, on the *
*-level the equation takes the form:

. (2.272)

If the system is closed over the channel (), it is closed over all overlying channels and in a general case, it is not closed over all underlying channels

Energy description of the levels , , and is expressed by

, (2.273)

, (2.274)

, (2.275)

. (2.276)

If the system is closed over the kinetic channel, i.e. , then energies

, , (2.277)

, , (2.278)

are conserved. If the system is open, motion-rest is also conserved but within the common bounds of the system and environment.

Copyright © L.G. Kreidik , 2001-2005