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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:





. (2.264b)

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)


, , (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.

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Copyright © L.G. Kreidik , 2001-2005