Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

i need to find the extremal of the functional $\int I(y,y'') dt$. Could anyone tell me the concepts of finding the extremal so that i can go about solving this one?

Update:

So my function was: G= $\int(ay+ $$\frac{1}{2}$$ b$y''^2$ )dt$ solving it using using the Euler–Lagrange equation as given in the link below, we get:

a+by''''=0

which on solving gives:

y=-(a(t^4))/(24b).

Am i right in solving this one?

share|improve this question
    
have you tried en.wikipedia.org/wiki/… (Euler-Lagrange equations - the link is to the case of higher derivatives, since you need it)? –  user8268 Apr 10 '11 at 13:50
    
for some reason my link doesn't go where it's supposed to, so go to the "Single function of single variable with higher derivatives" case –  user8268 Apr 10 '11 at 14:00
    
look up "calculus of variations" and "euler lagrange equations". there's also a cheap, easy to read dover book by gelfand and fomin entitled "calculus of variations" –  yoyo Apr 10 '11 at 14:52
    
@user8286 thank you, the link was very helpful!!! –  Siddhartha Apr 10 '11 at 19:40
add comment

2 Answers

Principally this should be related to the space on which the functional is defined. This depends on the exact form of $I$. However, if you forget these things, consider for $J(y)=\int I(y,y'')$, the expression $J(u+\tau v)$, where $\tau$ is a parameter, and $u,v$ are functions. Then differentiate the expression which respect to $\tau$, and put $\tau=0$. Rewrite the expression, using partial integrations (this only works if the space you decide to define the functional on is nice enough, for example compactly supported functions on the domain you are working on). You should then find some expression of the form $\int B(u,u'') v$. Since for a critical point this must be zero for all $v$, we conclude $B$ must be identically zero.

share|improve this answer
    
thank you for the explanation. But i did not get the last line, could you elaborate: B must be identically zero... –  Siddhartha Apr 10 '11 at 19:42
add comment

of course, the expression containing the terms that are unknown in the sense we trying a unknown rate of the function for the application with domain as the variable x. hence your answer in the predict of the Euler-Lagrange equation concept must needed the exact differentials in order to meet the bijection principle of a mapping.

ok, Asutosh sang K.M.P

share|improve this answer
    
I'm sorry, but I find this answer completely incomprehensible. –  Johannes Kloos Dec 6 '13 at 8:13
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.