- #1

sardel

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

[tex]\Phi(\gamma)=\int_a^b f(t)\cdot\gamma(t)dt,[/tex]

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

## Homework Equations

Linearity: [tex] L(\alpha\gamma(t)+\beta\gamma'(t))=\alpha L(\gamma(t))+\beta L(\gamma'(t))[/tex]

## The Attempt at a Solution

I can show that the functional is differentiable and equals its differential by showing linearity of [tex]\Phi[/tex].

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

[tex]\Phi(h)=0[/tex]

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

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