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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 5(36), Pages 60–72 (Mi vvgum131)  

Mathematics

Extremals of the equation for the potential energy functional

N. M. Poluboyarova

Volgograd State University

Abstract: To study the surfaces on the stability (or instability) is necessary to obtain the expression of the first and second functional variation. This article presents the first of the research of the functional of potential energy. We calculate the first variation of the potential energy functional. Proven some consequences of them. They help to build the extreme surface of rotation.
Let $M$ be an $n$ dimensional connected orientable manifold from the class $C^2$. We consider a hypersurface ${\mathcal M}=(M,u)$, obtained by a $C^2$ -immersion $u: M\to {\mathbf{R}}^{n+1}$. Let $\Omega\subset\mathbf{R}^{n+1}$ be a domain such that $\mathcal M\subset\partial\Omega;$ $\Phi$, $\Psi:  {\mathbf{R}}^{n+1}\to{\mathbf{R}}$$C^2$-smooth function. If $\xi$ the field of unit normals to the surface ${\mathcal M},$ then for any $C^2$-smooth surfaces ${\mathcal M}$ defined functional
$$ W({\mathcal M})=\int\limits_{\mathcal M}{\Phi(\xi)  d{\mathcal M}}+\int\limits_{\Omega}{\Psi(x)  d{x}}, $$
which we call the functional of potential energy. It is the main object of study.
Theorem of the first variation of the functional.
Theorem 3. If $W(t)=W({\mathcal M}_t),$ then
$$ W'(0)=\int \limits _{\mathcal M} {(div(D\Phi(\xi))^T-nH\Phi(\xi)+\Psi(x))h(x)  d{\mathcal M}}, $$
where $h(x)\in C^1_0(\mathcal M)$.
Theorem 4 is the the main theorem of of this article. It obtained the equations of extremals of the functional of potential energy.
Theorem 4. A surface $\mathcal M$ of class $C^2$ is extremal of functional of potential energy if and only if
$$ \sum \limits _{i=1}^{n}k_iG(E_i,E_i)=\Psi(x).$$

Corollary. If a extreme surface $\mathcal M$ is a plane, then the function $\Psi(x)=0.$
Theorem 5. If $f=x_{n+1}$ and $\Phi(\xi)=\Phi(\xi_{n+1}),$ then
$$\mathrm {div}((\xi_{n+1}\Phi'(\xi_{n+1})-\Phi(\xi_{n+1}))\nabla f)=\Psi(x)\xi_{n+1}.$$


Keywords: variation of functional, extreme surface, functional type area, volumetric power density functional, functional of potential energy, mean curvature of extreme surface.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-41-02479-р_поволжье_а


DOI: https://doi.org/10.15688/jvolsu1.2016.5.6

Full text: PDF file (403 kB)
References: PDF file   HTML file

UDC: 514.752, 514.764.274, 517.97
BBK: 22.15, 22.161

Citation: N. M. Poluboyarova, “Extremals of the equation for the potential energy functional”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 5(36), 60–72

Citation in format AMSBIB
\Bibitem{Pol16}
\by N.~M.~Poluboyarova
\paper Extremals of the equation for the potential energy functional
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2016
\issue 5(36)
\pages 60--72
\mathnet{http://mi.mathnet.ru/vvgum131}
\crossref{https://doi.org/10.15688/jvolsu1.2016.5.6}


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