# how to use variational principle to find the best value for parameter $\lambda$?

I need to minimize the following integral by varying parameter $\lambda$:

$$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$

The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy $f(0)=g(0,\lambda)$ and $f(\infty)=g(\infty,\lambda)=0$. And the integral is convergent.

What is the corresponding Eular-Lagrange equation?

Thanks- mike

• From what I understand, you use the EL equations to find an unknown function. If you know f and g and want to extremise the integral for $\lambda$ then this isn't a functional problem. – ClassicStyle May 20 '14 at 6:20

## 1 Answer

The "Euler-Lagrange" equation reads

$$\int\limits_0^{\infty}(f-g)\dfrac{dg}{d\lambda}dx = 0.$$

Knowing $f$ and $g$ you can (hopefully) compute this integral and then solve for $\lambda$.

Notice that if $\exists \lambda_0$ such that $f(x)=g(x,\lambda_0)$ then $\lambda_0$ leads necessarily to a true minimum of your integral, whereas the equation above may be a local one.