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.