INDIAN INSTITUTE OF SPACE SCIENCE AND TECHNOLOGY THIRUVANANTHAPURAM Submit ALL starred problems by 25 th March 2014.

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1 INDIAN INSTITUTE OF SPACE SCIENCE AND TECHNOLOGY THIRUVANANTHAPURAM B.Tech 4 th Semester(Aerospace/Avionics/ESS) Calculus of Variations - Assignment Submit ALL starred problems by 25 th March 24.. Solve the Euler-Lagrange equation for the functional / y (+x 2 y )dx subject to the end conditions y( ) = 9,y() =. 2. Derive Euler-Lagrange equation for the variational problem Deduce Beltrami identity from it. 3. Find the curve on which the functional has extremum value. 4. Find an extremal for the functional [y 2 y 2 ]dx which satisfies the boundary conditions y() = and y( π 2 ) =. F(x,y,y )dx, y( ) = y and y(x 2 ) = y 2. (y 2 +2xy)dx with y() =,y() = /2 5. Show that the Euler-Lagrange equation can also be written in the form F y F y x F y yy F y y y =. 6. It is required to determine the continuously differentiable function y(x) which minimizes the integral,y() =. ( + y 2 )dx, and satisfies the end conditions y() = (a) Obtain the relevant Euler equation, and show that the stationary function is y = x. (b) With y(x) = x and the special choice η(x) = x( x) and with the notation I(ǫ) = di(ǫ) F(x,y + ǫη(x),y + ǫη (x))dx, calculate I(ǫ) and verify directly that dǫ = when ǫ =. 7. Find the extremal of the following functionals (a) [ y 2 (y ) 2 2ycoshx ] dx, y( ) = y & y(x 2 ) = y 2

2 +y 2 (b) dx y 2 x2 +y 2 (c) dx x (d) (e) (f) (g) (h) /2 (xy +y 2 2y 2 y )dx, y() =,y() = 2 (y 2 +y 2 2ysinx)dx (y 2 y 2 +2xy)dx, y() =,y( π 2 ) = (y 2 +2xyy )dx; y( ) = y,y(x 2 ) = y 2 (i) (4ycosx y 2 +y 2 )dx; y() =,y(π) = x (y 2 +y 2 +2ye x )dx 8. Determine theshape of solidof revolution moving inaflow of gaswithleast resistance. y Direction of gas flow L x L (Hint : The total resistance experienced by the body is 4πρv 2 yy 3 dx where ρ is the density, v is the velocity of gas relative to the solid). 9. Prove the following facts by using COV: (a) The shortest distance between two points in a plane is a straight line. (b) The curve passing through two points on xy plane which when rotated about x axis giving a minimum surface area is a Catenary. (c) Thepathonwhichaparticleinabsenceoffrictionslidesfromonepointtoanother in the shortest time under the action of gravity is a Cycloid(Brachistochrone Problem). 2

3 . Find the extremal of the functional and subject to the constraint (y 2 y 2 )dx, y() =, y(π) = ydx =.. Find the extremal of the isoperimetric problem subject to 4 ydx = Determine y(x) for which,y() =. 4 y 2 dx, y() = 3,y(4) = 24 x 2 +y 2 dx is stationary subject to y 2 dx = 2, y() = 3. Find the extremal of I = y(π) =. y 2 dx subject to y 2 dx = and satisfying y() = 4. Given F(x,y,y ) = (y ) 2 +xy. Compute F and δf for x = x, y = x 2 and δy = ǫx n. 5. Find the extremals of the isopermetric problem given that x ydx = constant. 6. Prove the following facts by using COV: x y 2 dx (a) The geodesics on a sphere of radius a are its great circles. (b) The sphere is the solid figure of revolution which, for a given surface area has maximum volume. 7. If y is an extremizing function for then show that of δi = for the function y. F(x,y,y ),y( ) = y,and y(x 2 ) = y 2 8. Find y(x) for which and y( ) = y and y(x 2 ) = y 2. { } x δ ( y 2 x x 3)dx = 3

4 D 9. Write down the Euler-Lagrange equation for the following extremization problems (i) Extremize I(u,v) = F(x,y,u,v,u x,u y,v x,v y )dx dy where x,y are independent variables and u,v are dependent variables. D is a domain in xy plane and u and v are prescribed on the boundary of D. (ii) F(x,y,y (),y (2),...,y (m) dx x y(x ) = y, y( ) = y y (x ) = y, y ( ) = y... y (m ) (x ) = y (m ), y (m ) ( ) = y (m ) (iii) Max or Min F(x,y,y )dx where y is prescribed at the end points y( ) = y, y(x 2 ) = y 2, and y is also to satisfy the integral constraint condition J(y) = G(x,y,y )dx = k, where k is a prescribed constant. 2. Show that the extremals of the problem [p(x)y 2 q(x)y 2 ]dx where y( ) and y(x 2 ) are prescribed and y satisfies a constraint r(x)y 2 (x)dx =, are solutions of the differential equation d dx (pdy )+(q +λr)y = dx where λ is a constant. 2. Reduce the BVP d dx (xdy )+y = x, y() =, y() = dx into a variational problem and use Rayleigh-Ritz method to obtain an approximate solution in the form y(x) x+x( x)(c +c 2 x) 22. (Principle of least Action) A particle under the influence of a gravitational field moves on a path along which the kinetic energy is minimal. Using calculus of variation prove that the trajectory is parabolic. (Hint: Minimize I = 2 mv2 dt = 2 mvds = u 2 2gy +y 2 dx) where u is the initial speed. 23. Show that the curve which extremizes the functional /4 under the conditions y() =,y () =,y(π/4) = y (π/4) = 2 is y = sinx. (y 2 y 2 + x 2 )dx 24. Find a function y(x) such that y() = = y(π), y () = = y (π). y 2 dx = which makes (y ) 2 dx a minimum if 4

5 25. Find the extremals of the following functional 26. Find the extremals of the functional I(u,v) = 2xy +(y ) 2 dx where u and v are prescribed at the end points. x 2uv 2u 2 +u 2 v 2 dx 27. Find a function y(x) such that y() = = y(π), y () = = y (π) y 2 dx = which makes y 2 dx a minimum if ( ) π/2 2 ( ) 2 dx dy 28. Show that the functional 2xy + dt such that x() =, x(π/2) = dt dt, y() =, y(π/2) = is stationary for x = sint, y = sint. 29. Explain Rayleigh - Ritz method to find an approximate solution of the variational problem t t F(x,y,y )dx with prescribed end conditions y( ) = y & y(x 2 ) = y Solve the BVP y +y +x =, y() = y() = by Rayleigh - Ritz method. 3. Use Rayleigh - Ritz method to find an approximate solution of the problem y y +4xe x =, y () y() =, y ()+y() = e. ***END*** 5