Variational Calculus to minimize distance on a generalize 3D surface

[WackStr]

Homework Statement

A mountain is modeled by z=f(x,y) which is a known function.

a) What are the differential equations for x(t) and y(t) that minimize the distance between 2 points.

b) If
$$z=f(x,y)=(\sin^2 2\pi x)(\sin^2 2\pi y)$$
Solve the equations.

Homework Equations

The euler-lagrange equation.

The Attempt at a Solution

We have $$dz=dx\frac{dz}{dx}+dy\frac{dz}{dy}$$ so: $$ds=\sqrt{dx^2+dy^2+dz^2}$$
Using t as a parametrization we have
$$ds=\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}dt=L(x,y, \dot{x},\dot{y},t)dt$$
then the differential equations for x(t) and y(t) are obtained by:
$$\frac{\delta L}{\delta x}=0$$
$$\frac{\delta L}{\delta y}=0$$
However when trying to solve the equation I run into really long expressions that seem to be analyticall unsolvable. I think I'm missing a simple trick that needs to be used. A hint given is to note that the problem is symmetric under the exchange of x and y.

This is from hand and finch analytical mechanics Chapter 2 problem 11.

 

[mark726]

Dear student,

Thank you for your post. Solving this problem requires a few steps.

a) To minimize the distance between two points, we need to minimize the length of the curve connecting those two points. This can be done by minimizing the action functional:

S=\int_{t_1}^{t_2} L(x,y,\dot{x},\dot{y},t)dt

where L is the Lagrangian defined as:

L=\sqrt{\dot{x}^2+\dot{y}^2+\dot{z}^2}

To minimize this functional, we need to solve the Euler-Lagrange equations:

\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

In this case, the Lagrangian is independent of t, so the Euler-Lagrange equations reduce to:

\frac{\partial L}{\partial \dot{x}}=constant

\frac{\partial L}{\partial \dot{y}}=constant

b) To solve these equations, we need to substitute the given function z=f(x,y) into the Lagrangian and then use the symmetry property noted in the hint. This will simplify the equations and make them easier to solve. The solution will involve the use of elliptic integrals.

I hope this helps. Let me know if you need further clarification.