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Современные наукоемкие технологии
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ABOUT ONE INVERSE PROBLEM FOR CALCULUS OF VARIATIONS AND ITS APPLICATIONS

I. The initial value problem introduced by the author in works [1,2] is:

f   (I.1)

f,      (I.2)

f.          (I.3)

In this problem constant f and the functions f are determined as follows:

f

f

f;

where f and f are constants f and f is function

f

In the case f the problem (I.1) - (I.3) may be a problem with a point of singularity. The solution of this problem will simplify to a singularity point and can be determined by a generalized power series[1].

II. The inverse problem for calculus of variations [3] for the problem (I.1) - (I.3) is searching a functional with Lagrangian f, where the following are defined:

f.

The equation (I.1) must be the Euler - Lagrange equation as follows

f.

One from solutions of this problem is [1]:

f(II.1)

In last formula constants before argument t , variables f are defined from statement of a problem.

III. Applications. The formulas (I.1) - (I.3), (II.1) have next applications: curves with a point of return (a brachistochrone, Nail parabola), a dynamics problem with variable mass (the management of a movement of the rubbish collector in the space around the earth), the problem of finding an optimal shape of a body in hypersonic flow near a point of singularity [1].

References

  1. Svyatskov V.A. 2000. The equation of Euler - Lagrange in boundary layer with applications. Cheboksary, Chuvash State Pedagogical University, 165 p. (in Russian).
  2. Svyatskov V.A. One Metod of Calculation for Optimal Shape of a Body in Hypersonic Flow near a Singular Point.//High Speed Hydrodynamics. The Intetnational Summer Scientific School. - Russia, Cheboksary: 2002. - pp. 383 - 388.
  3. Bronshtein I.N. and Semendyaev K.A. The Handbook on Mathematics for Engineers and Students. - M.: Nauka. - 1986. - 544 p. (in Russian).

Библиографическая ссылка

Svyatskov V.A. ABOUT ONE INVERSE PROBLEM FOR CALCULUS OF VARIATIONS AND ITS APPLICATIONS // Современные наукоемкие технологии. – 2005. – № 8. – С. 31-32;
URL: http://www.top-technologies.ru/ru/article/view?id=23450 (дата обращения: 15.06.2021).

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