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Chapter 9: Q4P (page 481)

Change the independent variable to simplify the Euler equation, and then find a first integral of it.
$4 . {鈭�}_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$

Short Answer

Expert verified

Answer

The first integral of the Euler equation is $x' = \frac{C}{y \sqrt{y^{2} - C^{2}}}$ .

Step by step solution

01

Given Information

The given integral is ${鈭�}_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$ .

02

Definition of Euler equation

For the integral $I (蔚) = F {鈭�}_{x_{1}}^{x_{2}} (x, y, y') d x$ , the Euler equation is mathematically presented as $\frac{d}{d x} \frac{d F}{d y'} - \frac{d F}{d y} = 0$ .

03

Find the Euler equation of the given function

The given integral is ${鈭�}_{x_{1}}^{x_{2}} y \sqrt{y'^{2} + y^{2}} d x$ .

First, let鈥檚 change the variables as follows:

$\frac{鈭� F}{鈭� x'} = \frac{x' y^{3}}{1 + x'^{2} y^{2}} \frac{鈭� F}{鈭� x} = 0$

By using the two equations above, we can rewrite the integral.

$鈭� y \sqrt{y^{2} + y'^{2}} d x = 鈭� y \sqrt{y^{2} + y'^{2}} x' d y = 鈭� y \sqrt{y^{2} + x'^{- 2}} x' d y = 鈭� \sqrt{x'^{2} y^{2} + 1} d y$

Let $F (x, y, y') = y \sqrt{x'^{2} y^{2} + 1}$ .

By the definition of the Euler equation, $\frac{d}{d y} \frac{鈭� F}{鈭� x'} - \frac{蟺贵}{鈭� x} = 0$ .

Differentiate F with respect to x' and x.

$\frac{鈭� F}{鈭� x'} = \frac{x' y^{3}}{\sqrt{1 + x'^{2} y^{2}}} \frac{鈭� F}{鈭� x} = 0$

The Euler becomes as shown below.

$\frac{d}{d x} [\frac{x' y^{3}}{\sqrt{1 + x'^{2} y^{2}}}] = 0 \frac{x' y^{3}}{\sqrt{1 + x'^{2} y^{2}}} = C$

The differential equation is $x' = \frac{C}{y \sqrt{y^{4} - C^{2}}}$

Therefore, the first integral of the Euler equation is $x' = \frac{C}{y \sqrt{y^{4} - C^{2}}}$

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Most popular questions from this chapter

Write the胃Lagrange equation for a particle moving in a plane ifV=V(r) (that

is, a central force). Use the胃equation to show that:

(a) The angular momentum r脳mvis constant.

(b) The vector r sweeps out equal areas in equal times (Kepler鈥檚 second law).

Show that the geodesics on a circular cylinder (with elements parallel to the z axis) are helics $a z + b 胃 = c$ , where a,b,c are constants depending on the given endpoints.(Hint: Use cylindrical coordinates) Note that the equation $a z + b 胃 = c$ includes the circles $z =$ const.(for $b = 0$ ), straight lines $胃 =$ const.(for $a = 0$ ), and the special heclices $a z + b 胃 = 0$ .

For small vibrations, find the characteristic frequencies and the characteristic modes of vibration of the coupled pendulums shown. All motion takes place in a single vertical plane. Assume the spring is unstretched when both pendulums hang vertically and take the spring constant as $k = mg / l$ to simplify the algebra. Hints: Write the kinetic and potential energies in terms of the rectangular coordinates of the masses relative to their positions hanging at rest. Don鈥檛 forget the gravitational potential energies. Then write the rectangular coordinates $x$ and $y$ in terms of $胃$ and $蠒$ , and for small vibrations approximate $蝉颈苍胃 = 胃, 肠辞蝉胃 = 1 - 胃^{2} / 2$ , and similar equations for $蠒$ .

In the brachistochrone problem, show that if the particle is given an initial velocity $v_{0} 鈮� 0$ , the path of minimum time is still a cycloid.

Set up Lagrange鈥檚 equations in cylindrical coordinates for a particle of mass $m$ in a potential field $V (r, 胃, z)$ . Hint: $v = \frac{d s}{d t}$ ; writein cylindrical coordinates.

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