Minimising the action for surface area in AdS

[mattie_p]

Hello people.
I've got an action which needs minimising

$$\int dr \ r \sqrt{U'^{2}+U^{4}}$$

Where U(r). Simply plugging this into the EL equations yields a nasty looking 2nd order nonlinear differential equation. I'm just wondering if there's an easier way of solving for U(r). I've tried passing over into Hamiltonian mechanics but that seemed to confuse matters slightly (I probably got it wrong). Wondering if there's some implementation of Noether's Theorem that could give a solvable differential equation. As always, much thanks for your help.

 

[gammamcc]

I see integral w.r.t. tau and U is a function of what? r?

 

[gammamcc]

I see integral w.r.t. tau and U is a function of what? r?
BTW: What is the interval of integration?

 

[mattie_p]

Hey, thanks for having a look at this,
I think latex has just made the r look like a tau. It's meant to read
Integral dr*r*(U'^2 + U^4)^1/2

Also, the integration region is between 0 and some constant L, but that will lead to an inifinity, so the problem is actually integrated between 0 and (L^2-c^2)^1/2. I know the answer is roughly U(r)=(L^2-r^2)^-1/2, but my real problem is trying to derive this.

 

[gammamcc]

Well, it seems the integral may very well be zero with the right choice of U' and U over an interval of positive r. Considering the even powers involved, there are not many choices other than U= (?)




























ans: U=0 (identically).