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.