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Topic: What is Euler-Lagrange equation for this form?
Replies: 2   Last Post: Jun 9, 2012 1:25 PM

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RGVickson@shaw.ca

Posts: 1,645
Registered: 12/1/07
Re: What is Euler-Lagrange equation for this form?
Posted: Jun 9, 2012 1:25 PM
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On Friday, June 8, 2012 5:32:47 PM UTC-7, MBALOVER wrote:
> Hi all,
>
> I am not a math student. I am a software engineer and I am trying to
> program an algorithm which tries to solve an optimization problem. So
> please do not think this is a question about homework.
>
> I already looked at many Calculus of Variations textbooks and only
> find the Euler-Lagrange equations for this form: E = Integral
> f(x,y,x',y',s)ds where x and y are functions of the independent
> variable s. x', y' are 1st derivatives of x and y. E is the cost
> function to minimize.
>
> However my cost function has the form
>
> E = Integral f(x,y,x',y', x'', y'' ,s)ds. where x'' and y'' are 2nd-
> dertivatives
>
> Could you please let me know what are the Euler-Lagrange equations for
> my cost function? Or please point me a book that helps to derive the E-
> L equation of my form? Or at least some hints how to derive the E-L
> equation for this integral.
>
> Thank you.


Check out the free lecture notes in http://www.math.nps.navy.mil/~bneta/4311.pdf , especially from page 104 onwards. That gives the Euler equation for the general case of one y but several derivatives of y; doing it for two or more y-variables (and their several derivatives) would be an extension.

RGV



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