L. PETROVA 209

when the closed exterior forms is obtained from the un-

closed evolutionary form, and the identical relation is

realized. From such relation one can get the state func-

tional and find the state function. This fact will point to

the transition of material system into the locally equilib-

rium state.

The transition of the material system from nonequilib-

rium state into the locally-equilibrium one means that the

unmeasurable quantity described by the nonzero com-

mutator of the unclosed evolutionary differential form,

which acts as an internal force, transforms into the meas-

urable quantity. In material system, this reveals as the

emergence of certain observable formations, which de-

velop spontaneously. Such formations and their manifes-

tations are fluctuations, turbulent pulsations, waves, vor-

tices, and others [7].

It appears that the transition from the solutions of the

first type to the generalized solution corresponds to the

transition of material system from the nonequilibrium

state to the locally equilibrium one that is accompanied

by the emergence of a certain (observable) formation in

material system. The discontinuous functions that corre-

spond to generalized solutions just describe such obser-

vable formations.

Thus we obtain that the discrete realization of general-

ized solution points to the emergence of a certain (ob-

servable) formation in material system that is described

by discontinuous functions corresponding to generalized

solutions.

It may be also noted that the type of solutions to the

equations describing material systems is of great signifi-

cance for mechanics and physics of continuous media. In

mechanics and physics of continuous media the same

equations are considered (the equations of conservation

laws for energy, linear momentum, angular momentum,

and mass). The set of Navier-Stokes equations is an ex-

ample [5]. However, the approaches to solving these equa-

tions in mechanics and physics are different. In physics

the interest is expressed in only generalized solutions that

are invariant ones and describe measurable physical quan-

tities (but not the process itself), and noninvariant solu-

tions are ignored (even if they have a physical meaning).

The aim of mechanics of continuous media is to describe

the process of the continuous media evolution. And in

this case the numerical methods of solving differential

equations are commonly used without studying the inte-

grability conditions of these equations. The question of

searching for invariant solutions that are realized only

under additional conditions is commonly not posed. That

is, one considers the solutions that are not functions.

Such restricted approaches, both in physics and me-

chanics, lead to nonclosure of relevant theories and this

has some negative points. In mechanics without finding

the generalized solutions it is impossible to describe such

processes as the emergence of vorticity, turbulence and

others. The physical approach enables one to find allowed

invariant solutions, however, in this approach there is no

way to say at what time instant of evolutionary process

one or another exact solution was realized. This does not

also discloses the causality of phenomenon described by

these solutions. It is evident that in mechanics, as well in

physics, it is necessary to seek for solutions of both types.

In particular, in the case of gas-dynamic system such an

approach had been studied in paper [5].

3. On the Problem of Numerical Solving the

Differential Equations

As it was noted, the equations that describe actual proc-

esses are definded on manifolds (tangent, accompanying),

which are nonintegrable. If to model the equations on

such a manifold, one can obtain, without additional con-

ditions, the solutions of only first type, i.e. the solutions

that depend on the commutator with nonzero value caused

by inconsistency of derivatives or equations in the set of

equations. It should be emphasized once more that such

solutions have physical meaning, namely, they describe

the nonequilibrium state induced by the physical proc-

esses proceeded in the system. The generalized solutions,

which are functions and describe discrete formations,

cannot be obtained by modelling the equations only on

original manifold, since they are obtained on integrable

structures that do not belong to original nonintegrable

manifold. Therefore, to obtain the generalized solutions

by numerical simulation, one must use two systems of

reference. One more problem of obtaining the general-

ized solution relates to the fact that the integrable struc-

tures with generalized solution are not initially given, and

they are realized spontaneously in the process of inte-

grating under the realization of additional conditions,

namely, the integrability conditions. (As additional con-

ditions it may serve, for example, the characteristic con-

ditions, the dynamical conditions of the consistency of

equations in the set of equations [8] and so on). To obtain

the integrable structures, it is necessary to trace for the re-

alization of additional conditions, which define the inte-

grable structures, in the process of numerical integrating

the equations on the original manifold. This gives a pos-

sibility to obtain the instant of realization the generalized

solution.

In this case, the transitions from inexact solutions to

generalized ones describe the process of emergence of

any observable formations (in particular, such as waves,

vorticity and others), which intensity is definded by gen-

eralized solution.

As it was noted, in mechanics and physics the interest

is expressed in various types of the solutions to equations.

The methods of numerical solving the equations relate to

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