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Minimizing arc length

  1. Nov 26, 2005 #1
    The problem I am working on asks me to find the curve on the surface z=x^(3/2) which minimizes arc length and connects the points (0,0,0) and (1,1,1).
    Here's what I did:
    Integral [sqrt(dx^2+dy^2+dz^2)]
    Integral [dx sqrt (1+(dy/dx)^2 +(dz/dx)^2]
    Integral [dx sqrt (1 + (dy/dx)^2 + 9x/4)] since dz = 3/2 x^(1/2) dx

    Thus the "functional" is sqrt (1 + (dy/dx)^2 + 9x/4).

    Can I now take derivatives and substitute directly into the Euler-Lagrange equation and solve for y? Where/how do I apply the initial conditions -- that the endpoints are (0,0,0) and (1,1,1)?

    Am I on the right track with this one?
  2. jcsd
  3. Nov 26, 2005 #2

    Physics Monkey

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    Looks like you're doing ok to me. In this case the constants of integration are obtained from the boundary conditions i.e. y(0) = z(0) = 0 and y(1) = z(1) = 1.
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