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!

Calculus of variations changing variables

  1. Apr 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Hi
    I am given the functional

    x?S[y]=%5Cint_%7Ba%7D%5E%7Bb%7D%5Cfrac%7Bx%5E%7B3%7Dy%5E%7B%5Cprime2%7D%7D%7By%5E%7B4%7D%7D%20dx.png

    I am asked to show that if png.png and with an appropriate value for png.png that

    png.png

    2. Relevant equations



    3. The attempt at a solution
    So I get

    du%7D=%5Cfrac%7By%5E%7B%5Cprime%7D%28u%29%7D%7B%5Cbeta%20u%5E%7B%5Cbeta-1%7D%7D.png

    png.png
    png.png

    If I set 2.png then I get

    png.png

    I think that it is correct but what about the factor of 2?
     
  2. jcsd
  3. Apr 9, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Your work looks correct. Unless you do another variable transformation, you will not get rid of the 2.
     
  4. Apr 9, 2015 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I imagine that since you are trying to find stationary values of this integral, they are not affected by a constant multiple, so you can drop the factor. I think it should be -2, but that's a minor point.
     
  5. Apr 9, 2015 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    This would depend on the order of the integration limits (note that ##a## was the lower integration limit in ##x## but ##A## is the upper integration limit in ##u## - of course I am just making the arbitrary inference that ##a## corresponds to ##A## here ...).
     
  6. Apr 9, 2015 #5

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, didn't notice that.
     
  7. Apr 22, 2015 #6
    Thanks guys.
    I may have missed this in my notes PeroK but if we are trying to find stationary points of a functional constant multiples can be ignored?
    James
     
  8. Apr 22, 2015 #7

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Yes, but there is nothing strange about this. It works this way for functions as well, if ##f(x)## has the stationary point ##x=2##, then so does ##2f(x)##.
     
  9. Apr 22, 2015 #8
    Yes, thanks for the clarification.
    James
     
  10. Apr 22, 2015 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Positive multiples can be ignored, but omitting negative multiples changes the direction of optimization.
     
  11. Apr 22, 2015 #10

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    But it does not change the fact that the point is stationary. Just exchanges minima for maxima.
     
  12. Apr 22, 2015 #11

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Of course, but that is another issue. It certainly changes the second-order tests, so that instead of looking to see if the Hessian is positive-definite, we look instead to see if it is negative-definite. However, when I taught this stuff I recommended that students just "memorize" the conditions for a minimum, then switch the sign of the objective if the problem was a maximization; that eliminates the need for a whole raft of special cases, etc.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted