Minimizing Aphi' + Bcos(phi) in [0,L] w/ phi(0)=0

[dirk_mec1]

I'm trying to find a function for x in [0, L] that minimizes this:

$$\int_0^{L} A \phi(x) \frac{ d \phi(x) }{dx} + B cos(\phi(x))\ d\mbox{x}$$

For real (given) positve numbers A and B.

with
$\phi(0) = 0$
$\phi(x)$ is an increasing positve function.

Can somebody point me in the right direction?

 

[jfizzix]

Sounds like a calculus of variations problem.

I'm no expert, but the principal thing to do would be to find the Euler-Lagrange equation for $\phi(x)$. The solution (or solutions) to that equation will be the function which minimizes that integral.

I don't exactly know how to do this, but if you look at the Wikipedia article on the calculus of variations, and go down to the section on the Euler-Lagrange equation, you should find some useful tips there.

 

[dirk_mec1]

Yes I get the zero function then but that is not what I am looking for.

 

[jfizzix]

the Euler-Lagrange equation only has zero as a solution?

 

[dirk_mec1]

Yes I get the zero function then but that is not what I am looking for.

$$\frac { \partial L } { \partial \phi } = A \frac{d \phi}{dx} -C sin ( \phi(x))$$

$$\frac{d}{dx } \left( \frac { \partial L } { \partial \phi' } \right) = A \frac{d \phi}{dx}$$

$$\frac { \partial L } { \partial \phi } - \frac{d}{dx } \left( \frac { \partial L } { \partial \phi' } \right) = C sin ( \phi(x)) = 0$$

 

[jfizzix]

well that sure is interesting...

How do you know that the zero function isn't actually the solution you're looking for?

 

[jfizzix]

Also, since you're integrand doesn't depend explicitly on $x$, you can try the simplified Euler Lagrange equation instead:
$L-\frac{\partial L}{\partial \phi}\frac{d \phi}{d x} = const$.
Here, that would translate to
$A \phi \phi' + B Cos(\phi) -(A \phi' - B Sin(\phi))\phi'= const$
or rather
$A (\phi-\phi') \phi' + B \big(Cos(\phi) -\frac{d}{dx}( Cos(\phi)\big)= const$

I don't know how to solve this off the top of my head, but this equation probably admits more than zero as a solution.

 

[dirk_mec1]

Yes, you are right but I do not see how this is going to help me. This might be easier than an integral but this a non linear DE and I do not think an analytical solution exists.

 

[pasmith]

Since $\phi\phi' = \frac12 (\phi^2)'$ you are trying to minimize $$\frac{A\phi(L)^2}{2} + \int_0^L B\cos(\phi(x))\,dx.$$ The Euler-Lagrange equation is then $$B\sin(\phi) = 0$$ so that indeed $\phi(x) = 0$.

Do you have reason to believe that any other (differentiable) solutions exist? Also, although $\phi(x) = 0$ is not a strictly positive strictly increasing function, it is a positive increasing function.

(For weak solutions, you can take $$\phi(x) = \begin{cases} 0 & 0 \leq x < x_0 \\<br /> (2n+1)\pi & x_0 \leq x \leq L \end{cases}$$ for some positive integer $n$ and if $0 < 2L - \frac{A(2n+1)^2\pi^2}{2B} < L$ then for any $0 < x_0 < 2L - \frac{A(2n+1)^2\pi^2}{2B}$ you have $$\frac{A\phi(L)^2}{2} + \int_0^L B \cos(\phi(x))\,dx = \frac{A(2n+1)^2\pi^2}{2} - B(L - x_0) < BL,$$ the last being the value attained when $\phi$ is zero.)

 

[dirk_mec1]

I made a mistake with the model. I will create a new thread.