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Form Euler's equation for the integral $\iint \nu \, dS$, where $\nu$ is the segment of the normal between the surface and the plane $xy$.

Euler's equation for the integral

where $f$ takes given values on the contour, is capable of becoming an identity. Prove that a necessary and sufficient condition for this is the equality:

(On the other hand, a necessary and sufficient condition is the independence of the value of the integral from the shape of the surface $S$.)

Find Euler's equation for the problem of the minimum of the integral

for given values of $u$ on the contour of the region.

Find the minimum of the integral

if the values of the function $u$ are given on the contour. Here:

Find Euler's equation for the problem of the extremum of the integral

Obtain the differential equation of minimal surfaces, i.e., among all surfaces $z=z(x,y)$, resting on a given contour $C$ (or several contours), find the one having the smallest area. In particular, find the minimal surfaces whose equation has the form $z = \varphi (x)+\psi (y)$.

Show that if we have a minimal surface $z=z(x,y)$, resting on a given contour $C$, then this surface gives a strong minimum of the integral

Show that the problem of the minimum of the integral

where

leads to Euler's equation

where

(an equation of this kind arises from the problem of the plane steady adiabatic motion of a compressible fluid with velocity potential $\varphi$, if we set:

$\alpha$, $\gamma$, $U$ are constants).

Let $\xi (x,y)$, $\eta (x,y)$ be curvilinear coordinates, and let:

Starting from the first variation of the integral

show that the Laplace operator $\Delta u$ transforms to the new curvilinear coordinates by the formula

Example. Polar coordinates, where $ds^{2} = dr^{2} + r^{2}d\varphi^{2}$.

The values of the function $u$ are given on the boundary surface of the volume $V$.

Find the necessary condition for the minimum of the integral

The same question applied to the integral

Let $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ be curvilinear orthogonal coordinates, defined by the formulas:

Let also

Using the first variation of the integral

show that the Laplace operator $\Delta u$ in the new coordinates has the form

Example. Spherical coordinates, where $ds^{2} = dr^{2} + r^{2}\sin^{2}\theta \, d\varphi^{2} + r^{2}d\theta^{2}$.

Show that for the problem of the minimum of the integral

where

Euler's equations coincide with Maxwell's equations for the propagation of electromagnetic waves:

Form the differential equation of the longitudinal vibrations of a rod, starting from Hamilton's principle.

The rod is located along the axis $OX$ between the points $(0,0,0)$ and $(l,0,0)$ and works under longitudinal compression and tension. Notation: $u(x,t)$ --- displacement; $\omega (x)$ --- cross-sectional area; $\rho (x)$ --- density of the material; $E$ --- modulus of elasticity; $\sigma$ --- Poisson's ratio; $I(x)$ --- polar moment of the cross-section relative to the point $(x,0,0)$; $F(x)$ --- external force per unit mass. Determine what boundary conditions can be considered at the ends.

Form the differential equation of small transverse vibrations of a rod, starting from Hamilton's principle. The rod of length $l$ is located along the axis $OX$ and works under bending. The vibrations occur in the plane $XOY$. Notation: $v(x,t)$ --- displacement; $\omega (x)$ --- cross-section; $\rho (x)$ --- density of the material; $E$ --- modulus of elasticity; $I(x)$ --- moment of inertia of the cross-section relative to the axis passing through its center of gravity perpendicular to $XOY$; $F(x)$ --- external force, per unit mass. Also find the form of the limiting conditions that must be satisfied in various cases.

Starting from Hamilton's principle, form the differential equation of small transverse vibrations of a membrane, located in the plane $XOY$ and working only under tension. Notation: $\zeta (x,y,t)$ --- displacement, perpendicular to $XOY$; $\rho (x,y)$ --- surface density; $S_{0}$ --- area of the membrane in the equilibrium position; $P$ --- tension of the membrane, constant, giving a force acting in the plane $XOY$ per unit length of the contour bounding any element of area; $F(x,y)$ --- external force, perpendicular to $XOY$, per unit mass; $\varphi$ --- external force, perpendicular to $XOY$, acting on the contour, per unit length of the contour. Also find the form of the limiting condition on the contour $C$ of the membrane, when it is not fixed (if it is fixed, then the limiting condition will be $\zeta = \varphi (c,t)$ --- a given function of the position of the point on the contour).

Form the differential equation of equilibrium of a membrane and determine the boundary conditions in the case where an external force acts on it only on the contour, equal to $k(s)\zeta (s)$, where $k$ is a coefficient of proportionality, and $\zeta (s)$ is the transverse displacement of a point of the membrane. The force is directed opposite to the displacement.

Form the exact differential equation of equilibrium of a membrane serving as the bottom of a cylindrical vessel in which a liquid of density $\rho_{0}$, of depth $h$ (relative to the original position of the membrane), is poured. Notation: $\zeta (x,y,t)$ --- displacement of a point of the membrane in the vertical direction $OZ$ (positive direction downward), $P$ --- constant tension of the membrane.

Starting from Hamilton's principle, form the equation of small transverse vibrations of a thin plate. The plate works under tension and bending, its thickness is $2h$. In the equilibrium position the midplane of the plate lies in the plane $XOY$ and is bounded by the contour $C$. Here: $\zeta (x,y,t)$ --- displacement of a point along $OZ$, $\rho$ --- constant density of the plate, $E$ --- modulus of elasticity, $\sigma$ --- Poisson's ratio, there are no external forces. Determine the character of the limiting conditions.

Derive the equations of equilibrium of an elastic isotropic body, starting from the principle of virtual displacements, i.e., from the condition:

where

$\lambda$ and $\mu$ are elastic constants, $A$ is the strain energy per unit volume, $X$, $Y$, $Z$ are the components of the external force per unit mass, $u$, $v$, $w$ are the displacements of points of the elastic body, given on the surface; $u_{x}$ denotes $\frac{\partial u}{\partial x}$, and so on.

Derive the equations of motion of an elastic isotropic body, starting from Hamilton's principle:

where

and the quantity $\Pi$ has the same meaning as in problem 6098.