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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