Math Problems Vault

Problem 5961

Find sufficient conditions for a strong minimum of the integral

I = \int _{x_{1}}^{x_{2}} f(x, y) \, dx

given that the endpoints of the minimizing curve lie respectively on the lines $x = x_{1}$ and $x = x_{2}$ .

Examples:

$I = \int_{-1}^{+1} (y^{3} - 3x^{4}y) \, dx$ ;
$I = \int_{0}^{1} y(y+x) \, dx$ ;
$I = \int_{-a}^{+a} \left(x^{2}y + \frac{1}{3}y^{3} - a^{2}y\right) \, dx; \ a > 0$ ;
$I = \int_{0}^{x_{1}} (6ax - y^{2})y \, dx; \ a > 0, \ x_{1} > 0$ .

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Problem 5962

Find the minimum of the integral

I = \int _{(x_{1},y_{1})}^{(x_{2},y_{2})} (y^{2} + 2xyy') \, dx.

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Problem 5963

Find the extremum of the integral

I = \int _{(x_{1},y_{1})}^{(x_{2},y_{2})} (xy + y^{2} - 2y^{2} - 2y^{2}y') \, dx.

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Problem 5964

Find the extremum of the integral

I = \int _{(x_{1},y_{1})}^{(x_{2},y_{2})} (x^{2} + y^{2} + yy') \, dx.

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Problem 5965

Find the extremal of the integral

I = \int _{A}^{B} \frac{\sqrt{1+y'^{2}}}{y} \, dx,

passing through given points $A$ and $B$ , lying in the upper half-plane $y > 0$ .

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Problem 5966

Find the extremals of the integral

I = \int _{(x_{1},y_{1})}^{(x_{2},y_{2})} \sqrt{y(1+y'^{2})} \, dx, \quad y_{1} > 0, \quad y_{2} > 0.

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Problem 5967

Find in quadratures the extremals of the integral

I = \int _{M_{1}}^{M_{2}} y^{n} \sqrt{1+y'^{2}} \, dx.

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Problem 5968

Find the extremals of the integral

I = \int _{M_{1}}^{M_{2}} r^{n} \, ds

(in polar coordinates $r$ , $\theta$ ), expressing the $n$ -th moment of the arc $M_{1}M_{2}$ relative to the origin.

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Problem 5969

A material point moves in a plane so that its speed $v$ is inversely proportional to the distance to the origin, and $\int v \, ds$ , taken along the trajectory between any two of its points, is an extremum. Find the direction of the force acting on the point, and the orbit of the point.

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Problem 5970

A plane curve is the orbit of a material point moving freely under the action of a central force, and at the same time is a brachistochrone (see problem 5989) for a different motion, this time under a different central force. Show that if $v$ and $v'$ are the speeds of the point in these two motions at some point of the trajectory, then for all points of the trajectory the product $vv'$ remains constant.

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Problem 5971

Among all curves joining the origin to the line $x = a$ , determine the one which, together with the axis $Ox$ and the given line, encloses an area of given magnitude $S$ , and has the smallest length.

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Problem 5972

On the sides $OA$ and $OB$ of an angle with vertex $O$ are given points $A$ and $B$ ; it is required to join them by the shortest line $ACB$ , under the condition that the area $OACBO$ has a given magnitude $S$ .

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Problem 5973

Find the surface of revolution about the axis $OX$ , having a given area, and enclosing the largest volume, given that the surface intersects the axis of revolution exactly twice.

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Problem 5974

Find all integrals

I = \int _{(x_{1},y_{1})}^{(x_{2},y_{2})} F(x, y, y') \, dx,

for which straight lines serve as extremals.

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Problem 5975

Find all integrals

I = \int _{x_{1}}^{x_{2}} F(x, y, y') \, dx,

for which the extremals are the parabolas $y = ax + bx^{2}$ .

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