410HW4 - Math 410 Homework Solutions February 23, 2009 1...

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Unformatted text preview: Math 410 Homework Solutions February 23, 2009 1 Homework #4 1. Let S be a surface of revolution about the z-axis. It is defined by an equation of the form x 2 + y 2 = f ( z ). Let be a geodesic on S and let u ( t ) = ( x ( t ) ,y ( t ) ,z ( t )) be its parametrization by arc length. (a) Derive the Euler-Lagrange equations for x,y, and z . Solution Consider the following functional F = Z | u ( t ) | + ( t )( x 2 + y 2- f ( z )) dt For to be a geodesic, u must satisfy the Euler equations for this functional, i.e. 2 x- d dt x | u | = 0 2 y- d dt y | u | = 0- df dz- d dt z | u | = 0 Using the fact that u is parameterized by arc length, we get 2 x = x 00 2 y = y 00- df dz = z 00 (b) Show that xy- yx = c 1 , where c 1 is a constant. Solution Multiplying the first equation by y, second by x, we find yx 00 = xy 00 . But notice d dt ( xy- yx ) = xy 00 + x y- y x- yx 00 = xy 00- yx 00 Therefore, d dt ( xy- yx ) = 0, i.e. xy- yx = C 1 . 1 (c) Show that the element of arc length in R 3 is given by ds 2 = dr 2 + r 2 d 2 + dz 2 . Solution First calculate that dx = cos dr- r sin d dy = sin dr + r cos d. Then ds 2 = dx 2 + dy 2 + dz 2 = cos 2 dr 2 + r 2 sin 2 d + sin 2 dr 2 + r 2 cos 2 d = dr 2 + r 2 d 2 + dz 2 . (d) Express the parametrization for the geodesic in cylindrical coordi- nates u ( t ) = ( r ( t ) , ( t ) ,z ( t )), and find the Euler-Lagrange equations for r,, and z ....
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410HW4 - Math 410 Homework Solutions February 23, 2009 1...

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