Stationary points of functional

[sardel]

Hello guys.

This is my first post at physics forums, so please be gentle :)

I am trying to understand functionals, so I am solving as many exercises from these lecture notes that I downloaded.

Homework Statement

Let $$f:[a,b]\rightarrow\mathbb{R}^k$$ be a continuous function and define

$$\Phi(\gamma)=\int_a^b f(t)\cdot\gamma(t)dt,$$

where $$\gamma:[a,b]\rightarrow\mathbb{R}^k$$ is parametrized curve. Show that $$\Phi$$ is differentiable and equals its differential. Determine the stationary points of $$\Phi$$.

Homework Equations

Linearity: $$L(\alpha\gamma(t)+\beta\gamma'(t))=\alpha L(\gamma(t))+\beta L(\gamma'(t))$$

The Attempt at a Solution

I can show that the functional is differentiable and equals its differential by showing linearity of $$\Phi$$.

However, I don't know how to find the stationary points of $$\Phi$$. From what I understand, I have to find all $$\gamma\in C^1([a,b];\mathbb{R}^k)$$ such that

$$\Phi(h)=0$$

for all $$h\in C^1_{0,0}([a,b];\mathbb{R}^k)$$.

I have no idea how to do this, as I can't see how to use something like the Euler-Lagrange equations. Any help is appreciated.

Thanks in advance,
Sardel

 

[sardel]

No love for functionals?

I should probably say that the definition of stationary points given in the lecture notes is

If $$\Phi$$ is a differential functional on $$D_{\Phi}=C^1([a,b];\mathbb{R}^k)$$ we call $$\gamma\in D_{\Phi}$$ a stationary point of $$\Phi$$ if the differential $$d\Phi_{\gamma}$$ vanishes on $$C^1_{0,0}([a,b];\mathbb{R}^k)$$, i.e. $$d\Phi_{\gamma}(h)=0$$ for all $$h\in C^1_{0,0}([a,b];\mathbb{R}^k)$$.

Further, $$C^1_{x,y}([a,b];\mathbb{R}^k)=\{\gamma\in C^1([a,b];\mathbb{R}^k)\mid \gamma(a)=x,\,\gamma(b)=y\}$$.

My problem is that the differential equals the functional itself, so $$d\Phi_{\gamma}(h)=\Phi(h)$$, which does not depend on $$\gamma$$. Am I missing something?

Thanks,
Sardel

 

[micromass]

Can you give me the link to the lecture notes you download? I promess to have a look, but I don't understand many of the notations you're using...

 

[sardel]

The lecture notes are available here: http://www.math.ku.dk/~solovej/MATFYS/MatFys2.pdf" . All the other chapters are available as MatFys1.pdf through MatFys6.pdf, and the directory listing can be viewed for the directory /MATFYS/

The exercise I am trying to solve is Exercise 2.6 and everything relevant should be in chapter 2.

 

[sardel]

The relevant chapter is available at www.math.ku.dk/~solovej/MATFYS/MatFys2.pdf

 

[micromass]

The Euler-Langrange equations are not directly applicable since $$\gamma$$ is maybe not C². But I think you can apply the proof of the Euler-Lagrange equations.

If $$\Phi(h)=0$$ for every $$h\in C_{0,0}$$. Then $$\Phi(h_1,0,0,...,0)=0$$. Now apply the fundamental lemma of calculus...

 

[sardel]

Thank you.

By doing this, I end up with $$\gamma_1(t)=0$$ for all $$t\in[a,b]$$ and analogously for $$\gamma_2,\ldots\gamma_k$$. Making $$(0,\ldots,0)$$ the only stationary point of $$\Phi$$. Is this correct?

 

[sardel]

Oops... Actually, I think I get $$f_1(t)=0$$ for all $$t\in [a,b]$$. But I want to conclude somthing about $$\gamma$$, right? I think there is something fundamentally wrong with the way I am thinking about this.

 

[micromass]

No, there is nothing wrong. You indeed end up with f=0. Thus we can split up in cases. If f=0, then every point is stationary. If f is not 0, then no point is stationary...

 

[sardel]

Ah ok, thank you very much.