Using E-L equations to get through a mountain range

[SteDolan]

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:

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

Which I can solve using the Euler–Lagrange equations.

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

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

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 \dot{y} to get \dot{y}(x,y) 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.

 

[pasmith]

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:

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

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

(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
<br /> \int_0^1 |\dot h|\,\mathrm{d}t<br />
which minimizes total change in altitude. By the chain rule
<br /> \dot h = \frac{\partial h}{\partial x} \dot x + \frac{\partial h}{\partial y} \dot y<br />
so you can take
<br /> 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}
and the Euler-lagrange equations are
<br /> \frac{d}{dt}\left(\frac{\partial L}{\partial \dot x}\right) - \frac{\partial L}{\partial x} = 0 \\<br /> \frac{d}{dt}\left(\frac{\partial L}{\partial \dot y}\right) - \frac{\partial L}{\partial y} = 0<br />

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