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
$$
W'(0)=\int \limits _{\mathcal M} {({\rm div}(D\Phi(\xi))^T-nH\Phi(\xi)+\Psi(x))h(x)\, d{\mathcal M}}, $$
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 $$\mathrm {div}((\xi_{n+1}\Phi'(\xi_{n+1})-\Phi(\xi_{n+1}))\nabla f)=\Psi(x)\xi_{n+1}.$$
