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32цессов, происходящих в многокомпонентных средах, тектонических процессов в земных недрах и других.
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APPLICATION OF THE VARIATION PRINCIPLE IN MECHANICS
Zhadan S.A.,
Master's student
Korkyt Ata Kyzylorda state University Kyzylorda, Kazakhstan
Seitmuratov A., Doktor of Physical and Matematical Sciences Korkyt Ata Kyzylorda state University, Kyzylorda, Kazakhstan
Zhamkeeva G.S.
Teacher researcher №15 school-lyceum named after M. Duissenova In Kyzylorda. Kazakhstan
Abstract
Students usually solve a variational problem in order to find a function that satisfies the stationary condition of a given function (within the framework of calculating variations - the equation of a function), that is, a function with (infinitesimal) stresses. The variational problem is also called the closely related problem of finding a function that reaches a given functional local extremum (functional equation) (in most cases, this problem is reduced to the first, sometimes almost complete). Usually this use of terms means that the problem is solved by calculating variations. Typical examples of variational problems are isoperimetric problems of geometry and mechanics; in physics - the problem of finding the equations of a field for a given type of activity for this field.
Keywords: functional, variational problems, variational principles, Newton's law, Ritz method, coordinate functions.
Introduction
In addition to the problems that require the determination of the extreme values of a particular function, there is a need to determine the extreme values of a particular quantity called functional in the
natural sciences. Functional is a variable whose values are determined by selecting one or more functions.
Examples of functionalities are quantities used in the science of mechanics, such as moments of inertia, static moments, and the coordinates of the center of gravity of a curve or surface in space. The science of
variational calculus studies methods for finding the extremums of functionalities (min or max). Problems that require min or max-analysis of alfunctionals are called variational problems.
Variational problems arise from the laws that require that the quantities characterized by a particular function be min or max in many processes studied in the mechanical and physical sciences. The laws thus described are called the principles of variation. Variational principles include: the law of conservation of energy, the law of conservation of momentum, the law of conservation of momentum, etc. In addition, the principles of variation - the Fermat principle in optics, the Castiliano principle in the theory of elasticity, etc. can be called.
Main part
One of the basic principles of variation in mechanics is the Ostrogradsky-Hamiltonian principle. According to this principle, of all the possible movements of the system of points of matter, which allow the connections, in fact, only the movement that gives a stationary value to the integral.
ihr-u)dt
•■■2 ■ z2 )
Kinetic energy of the system
T = 1 Em (x + y2 + z 2
2 i=\
determined by the formula and U the potential energy of the system is equal to. System j£(T - U)dt of Euler equations for a given integral is written as.
T-0
8x dt dxt
T-o
Syt dt Syt ' dz, dt SZ,
or
miXi - Fx = 0, miyi - Fy = 0 mZi - F.z = 0
From the obtained system of equations we see the differential equation of motion of material points. This is a known Newton's law.
Until now, the extreme values of the functionality have been sought from any possible curves. The found extreme curve was the exact solution of the variational problem. Since the direct methods search for an approximate solution of the variational problem, the Ritz method also seeks an approximate solution that seeks an exact solution in infinity. In the Ritz method, the approximate solution of the variational problem is sought from a linear W (x), W2 (x),... , Wn (x),... combination formed by constant n coefficients of the
first ai term of the phrase y = X aWWt (x)
i=1
functions, which are called coordinate functions and require selection. Of course, the
W (x), W (x),... , Wn (x),... functions must be
possible functions for a given variational problem, in other words, they must be subject to certain conditions. Thus, in the selected linear combination, the v[y(x)]
, X function of the
function becomes a a , X2
(p{aY,
a
a.
)
coefficients.
These
Jfo
Therefore, for this motion, the variation of this function must be zero. Here
T - the kinetic energy of the system, and U - the potential energy. Give the masses^ (i = 1,2 — n) and xi,yi,zi coordinates of each point in the system of material points and give it the force potential Suppose that the forces determined by It is necessary to create a differential equation of motion of the system.
p _ 8u p _ 8u ^ _ 8u
'x 8x/ 8y/ 8z
= 0
d, a, ...„a coefficients are determined by the
\ 7 2 7 7 n J
condition that the (p(&Y , d2,..., dn )function reaches the extremum, so they must be determined from
the system of ~— = 0 (i = \, 2,..., n) equations.
8ai
By moving to the limit state n —> to , the exact
to
y = (x) solution of the given variational
i=\
problem is obtained.
If we do not strive for a limit state, then we get an approximate solution of the given variational problem
8(
by solving a system of equations t—
(i = \, 2,..., n).
If this method searches for the approximate absolute minimum of the variance problem, then the minimum is obtained in excess. This is because the minimum along the possible lines is always equal to or less than that value. On the contrary, the maximum is missing.
Now let's look at the possible conditions of the
n
yn (x) combination. Of course, we must
i=\
not forget the condition that the possible curves are continuous and monotonous. Most importantly, they must meet the border conditions.
If the boundary conditions are given in the form of a homogeneous y(x0 ) = y(xx) = 0, or, in general, in
the form of A)+ A(x;) = 0 (j = 0,l),
then it is easier to give the coordinate functions in a way that satisfies these boundary conditions. The ^ coefficients here are constant. At the same time, it is
clear that the y aWt (x) function must
t=1
satisfy the boundary conditions for any at.
For example, if the boundary condition is given in
the form y(xo ) = y(x ) = 0, you
can select Wi (x) = (x - xo )(x - (x)
functions W,(x)= sin k*(x~ Xo ) (k = 1,2,...),
X1 xo
functions, or any continuous function that meets the W (xo ) =W (x ) = 0 conditions as coordinate functions.
However, if the boundary conditions are not homogeneous, for example, in the form y (x0 ) = y0, y(x^ ) = y, then it is better to look for the solution of the variational problem in the form
n
(x) + W (x). Here, the W0 (x)
yn =
i=1
function must satisfy the boundary W (x0 ) = y0, W (x ) = y conditions, and all other W (x) functions must satisfy the corresponding homogeneous boundary conditions, in particular the W (x0 ) =W i (xj ) = 0 conditions. When the coordinate functions are selected in this way, it is clear that the Wt (x) functions satisfy boundary conditions
for a. any yn (x). For example, you can take a linear W (x) function as a
W0 (x) = y—— (x - x0 ) + yo function.
xi - x0
In general, it is not easy to find a solution to the
- n t \
system of equations t— = 0 (t = 12,..., n). To
dat
simplify the problem, it is better to consider the dp
da
0 ( = 1, 2,..., n) function, which is
problems. For example, if, boundary conditions are
d2 z _ d2 z
given in a certain
dxz 8y2
f (x, y )
zone.
If it is necessary to find a solution to the Poisson equation, then this problem can be replaced by a variational problem to find the extremum of the functional. In this case, this equation is the Ostrogradsky equation, and its function is written as:
if
Y
+
f dz^2
_cx ) i ay
+
2zf (x, y )
dxdy.
quadraticly dependent on the unknown function and its
derivatives. In this case, the a equations are obtained
linearly with respect to the coefficients.
Successful selection of a combination of
W (x), W2 (x),... , Wn (x),... functions, called
coordinate functions, can make subsequent calculations easier.
This applies to the functionality x^, x2,..., x of the functions of the W variables, or to the
v[z(x:, x2,..., xn)] functionality that depends on
several functions.
The Ritz method is often used to search for exact or approximate solutions to mathematical physics
The function that gives an extremum to this function can be determined by any of the direct methods.
Theoretically, the probability of an approximate solution to the exact solution is determined by the transition to the limit state (n —> to ). The accuracy of the method can be found in special studies by Mikhlin, Kantorovich or Krylov.
In applied problems, in most cases, it is not necessary to find an exact solution to the variational problem. A simple method is used to assess the accuracy of the solution determined by the Ritz method, or any direct method. This method is easy to use, although theoretically less accurate.
For example, after comparing the values of the next yn (x) and yn+x (x) solutions identified during the search for the desired solution at several specially selected points between [xo, X ], if their absolute difference satisfies the accuracy condition, it leaves the yn (x) solution. Otherwise, repeat this test for
y«+i(x) and y„+2 (x). This accuracy checking
process continues until the required accuracy is reached.
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