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Finding the functional extremum

  1. Dec 20, 2014 #1
    1. The problem statement, all variables and given/known data
    I have been given a functional
    $$S[x(t)]= \int_0^T \Big[ \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)\Big] dt$$
    I need a curve satisfying x(o)=0 and x(T)=1,
    which makes S[x(t)] an extremum

    2. Relevant equations

    Now I know about action being
    $$S[x(t)]= \int_t^{t'} L(\dot x, x) dt$$
    and in this equation $$ L= \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)$$
    3. The attempt at a solution
    Is there any other way I can express Lagrangian to fit in this equation and hence I can do the integral? and is there any general solution for the lagrangian?
  2. jcsd
  3. Dec 20, 2014 #2


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    Why do you want to do the integral? What is wrong with using the Euler-Lagrange equations? This will give you a differential equation that your solution must satisfy to be an extremum the functional.
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