Extremizing functionals (Calculus of variations)

[phymatast]

Homework Statement

Find the curve y(x) that extremizes the functional J[y]= int({1-y'^2}/y,x=a..b) if the end points lie on two non-intersecting circles in the upper half-plane.

Homework Equations

Euler's equation: if F=F(x,y,y') then Euler's equation extremization is found from solving the ODE: F_y - d/dx(F_y') = 0 (where F_y denotes the derivative of F wrt y).

Since F is independant from x the problem reduces to solving y'*F_y' -F= c

The Attempt at a Solution

I used the Euler's equation to solve the given functional.

At some point I get: y'^2 = c*y -1
so if i want to solve for y' I will have to choose wether to give it + or - sign (first confusion).

Suppose the first problem is trivial. I got the extremum to be a parabola with 2 constants.


Now my main problem is to use the boundary conditions to determine the curve y(x).
So y(a) will be varying on the first circle and y(b) on the second.
I tried to read in my text (Gelfand and Fomin) but I couldn't just solve it.

Any help :)?

Thanks,
Joe

 

[mighty2000]

Dear Joe,

Thank you for your post and for sharing your progress so far on solving the given functional. It seems like you have a good understanding of the basic principles involved in finding the extremum of a functional using Euler's equation.

Regarding your first confusion about choosing the sign for y', it is important to remember that the sign of y' does not affect the overall shape of the curve, but rather the direction in which the curve is traced. In this case, the given functional is symmetric about the x-axis, so the sign of y' will not have a significant impact on the solution.

As for your main problem of using the boundary conditions to determine the curve y(x), you are on the right track. The boundary conditions will help to determine the specific values of the constants in the parabola that you have found. This can be done by substituting the values of y(a) and y(b) into the equation for y(x) and solving for the constants.

If you are still having trouble, I would recommend looking into the method of variation of parameters, which can be used to solve boundary value problems involving non-linear differential equations like the one you are working with. Additionally, seeking help from a professor or a tutor may also be beneficial in guiding you towards a solution.

Best of luck with your work!