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!

Action integral

  1. Apr 19, 2005 #1
    If you have an action integral

    \int_{A}^{B} \sqrt{F\mathbf{(r)}} dr


    F=a-bz^2 , b>0, a-bd^2>0

    the minimum of the action integral is equivalent to



    or am i doing this in a completeley wrong way?
  2. jcsd
  3. Apr 19, 2005 #2
    You jump around a lot with your notation. See if this is clear:

    If you have an Integral of this form(where y' is the derivative of y with respect to x):

    [tex] \Int F[y , y' , x ] dx [/tex]

    Then the function y which minimizes the integral is the solution to this differential equation:

    [tex] \frac {d}{dx} (\frac {\partial F}{\partial y'}) - \frac{\partial F}{\partial y} = 0 [/tex]
  4. Apr 19, 2005 #3


    User Avatar
    Science Advisor
    Homework Helper


  5. Apr 20, 2005 #4
    It was those equations i was trying to use...but probably in a wrong way.
    To explain my notation i give you the problem:

    A light ray's path in a medium with variating dielectricity constant


    between the two points A and B, minimizes, according to Fermat's principle, the action integral

    [tex]\int_{A}^{B} \sqrt{\epsilon(\vec{r})} |d\vec{r}| [/tex]

    A plane piece of glass with thickness 2d has a dielectricity function which, when the piece is in |z|<=d in a cartesian coordinatesystem, can be written

    \epsilon(\vec{r})=a-bz^2 ,b>0, a-bd^2>0


    Calculate the general form for the light ray's path in the piece of glass.


    The minimum of the action integral is equivalent to
    (But this is probably wrong)

    [tex]\frac{d}{dt}\frac{d(\sqrt{\epsilon(\vec{r})} )}{\dot{z}}-\frac{d(\sqrt{\epsilon(\vec{r})} )}{z}=0[/tex]

    and with this you get

    -1/2(a-bz^2)^(-1/2) * 2bz = 0

    and this must be wrong!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Action integral
  1. Group Action (Replies: 14)

  2. Reflex action (Replies: 7)

  3. Capillary action (Replies: 1)

  4. Capillary Action (Replies: 5)