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
$$\max\{h(x(t),y(t))\}$$
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
$$J[x,y] =\int_0^1 h(x(t),y(t))\,dt$$

(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
$$\int_0^1 |\dot h|\,\mathrm{d}t$$
which minimizes total change in altitude. By the chain rule
$$\dot h = \frac{\partial h}{\partial x} \dot x + \frac{\partial h}{\partial y} \dot y$$
so you can take
$$L(x, \dot x, y, \dot y) = \sqrt{\left( \frac{\partial h}{\partial x}(x,y) \dot x + \frac{\partial h}{\partial y}(x,y) \dot y \right)^2}$$
and the Euler-lagrange equations are
$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot x}\right) - \frac{\partial L}{\partial x} = 0 \\ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot y}\right) - \frac{\partial L}{\partial y} = 0$$

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