1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simple calculus - interpretation Euler-Lagrange equation

  1. Sep 2, 2006 #1
    This is not a homework question but one that is part of the course material and I can't really move on until I understand the basic calculus.

    I have a problem interpreting "d by dx of partial dF by dy' equals partial d by dy' of dF by dx" in the following question, which I set out and then highlight my problem.

    For F = (x^2 + y'^2)^1/2, find;

    partial dF/dx = x(x^2 + y'^2)^-1/2

    partial dF/dy = 0

    partial dF/dy' = y'(x^2 + y'^2)^-1/2

    dF/dx = p.dF/dx + p.dF/dy.y' + p.dF/dy'.y''


    Show that: d/dx(p.dF/dy') = p.d/dy'(dF/dx)

    Taking the RHS I believe is:

    p.d/dy'(dF/dx) = p.d/dy'[x(x^2 + y'^2)^-1/2 + p.dF/dy.y' +

    y'(x^2 + y'^2)^-1/2 y'']

    ditto = - xy'(x^2 + y'^2)^3/2 + p.d/dy' pdF/dy.y'

    - y'^2 y''(x^2 + y'^2)^3/2

    My problem is that I have forgotten how to interprete p.d/dy' pdF/dy.y' I know that the answer is y''(x^2 + y'^2)^-1/2 but I don't understand how one gets this answer when partial dF/dy = 0

    and I don't know how to interpret the LHS - d/dx(partial dF/dy') at all although I appreciate that it gives the same answer for this function as the RHS.
    I would appreciate some help on the fundamentals. Many thanks.
     
  2. jcsd
  3. Sep 2, 2006 #2

    StatusX

    User Avatar
    Homework Helper

    In this case [itex]\partial F/\partial y =0 [/itex], so that term drops out. In general you would write out that term like the others and do just what you did with them.

    The following:

    [tex]\frac{d}{dx} \left( \frac{\partial F}{\partial y'} \right)[/tex]

    is computed the same way as [itex]dF/dx[/itex]. Remember that [itex]\partial F/\partial y' [/itex] is a function you know, just like F. Just write it out like your fifth line above.
     
  4. Sep 7, 2006 #3
    Thank you StatusX. I've got it at last.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Simple calculus - interpretation Euler-Lagrange equation
Loading...