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Using E-L equations to get through a mountain range

  1. Sep 23, 2013 #1
    I'm just trying to get my head around calculus of variations so I gave myself a simple example to work with:

    Suppose I want to get between two points (a and b) through a mountain range with height described by the scalar field h(x,y) whilst staying as low as possible. I want to know what shaped path I should take.

    So I start by saying I want to minimise:

    [itex]\int h(x,y) ds = \int h(x,y) \sqrt{1 + \dot{y^2}} [/itex]

    Which I can solve using the Euler–Lagrange equations.

    Noting that h and ds are not explicitly dependent on [itex]\dot{y}[/itex] and y respectively I think the E-L equation can reduce to:

    [itex]\frac{\dot{y}}{1 + \dot{y^2}} \frac{dh(x,y)}{dx} = \frac{dh(x,y)}{dy} [/itex]

    Assuming some form of h(x,y), my difficulty is going from here to my path through the mountain range, y(x). I can solve the following quadratic in [itex]\dot{y}[/itex] to get [itex]\dot{y}(x,y)[/itex] but I'm unsure about the following integral to get to y(x). At first I thought I could ignore the y in the h(x,y) when i'm integrating with respect to x but since I'm after the path y(x) I can hardly say they are independent.

    I just wanted to check that I've had the right idea up to this point and that I can't use any arguments to justify ignoring the y in h(x,y) when i've got ∫f(h(x,y))dx, meaning I'm stuck with a horrible implicit integration by parts.

    Thanks for any help :-) Sorry my post is a little convoluted.
     
  2. jcsd
  3. Sep 23, 2013 #2

    pasmith

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    Homework Helper

    That's not the problem you think you are solving. You want to find the path which minimizes
    [tex]
    \max\{h(x(t),y(t))\}
    [/tex]
    over all continuous paths [itex](x(t),y(t))[/itex] between given end points. That path might not be expressible in the form [itex]y = f(x)[/itex], and in any case you won't necessarily find it by applying variational principles to the functional
    [tex]
    J[x,y] =\int_0^1 h(x(t),y(t))\,dt
    [/tex]

    (Parametrizing by arclength is a bad idea in variational problems, since the length of the path (and hence the limits of the integral) will change determine on the path taken.)

    A problem which is in principle solvable by using EL is that of minimizing
    [tex]
    \int_0^1 |\dot h|\,\mathrm{d}t
    [/tex]
    which minimizes total change in altitude. By the chain rule
    [tex]
    \dot h = \frac{\partial h}{\partial x} \dot x + \frac{\partial h}{\partial y} \dot y
    [/tex]
    so you can take
    [tex]
    L(x, \dot x, y, \dot y) = \sqrt{\left( \frac{\partial h}{\partial x}(x,y) \dot x + \frac{\partial h}{\partial y}(x,y) \dot y \right)^2}[/tex]
    and the Euler-lagrange equations are
    [tex]
    \frac{d}{dt}\left(\frac{\partial L}{\partial \dot x}\right) - \frac{\partial L}{\partial x} = 0 \\
    \frac{d}{dt}\left(\frac{\partial L}{\partial \dot y}\right) - \frac{\partial L}{\partial y} = 0
    [/tex]

    Note that [itex]L[/itex] can depend expressly on [itex]x[/itex] and [itex]y[/itex] through the derivatives of [itex]h[/itex].
     
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