# Finding the first integral for equations of motion

1. May 20, 2014

### Elodin

Hi,
I'm trying to find the first integral of motion for this set of PDE. I'm not really that familiar with this method of solving PDE.
I'd really appreciated if someone could help me.

The problem seems quite complicated but then again the paper I've got it from says it's a straigthforward calculation. (Getting from equation (23) to (24) in this paper: http://arxiv.org/pdf/hep-th/9803135v2.pdf)

$$\begin{eqnarray} \left(\frac{\frac{H}{G}}{\sqrt{U'^2+\frac{H}{G}}}\right)' & = & 0 \\ \left({U'\over\sqrt{U'^2+{H\over G}}}\right)' & = & \frac{\partial_U (\frac{H}{ G})}{\sqrt{U'^2+\frac{H}{ G}}}. \end{eqnarray}$$

$$\mbox{Where H and G are functions of U; and U and X are functions of } \sigma. \hspace{8pt}' \mbox{ denotes a derivative to } \sigma. \mbox{H and G are given by}:$$

$$\begin{eqnarray} H = 1 - \frac{U_0^4}{U^4} \\ G = \frac{g^2_{eff}} {U^4} \end{eqnarray}$$

$$\mbox{I know the solution is: } \left(\frac{G}{H} \right)^2 U^{'2} + {G \over H} = \mbox{constant}.$$

I don't think exact knowledge of what H and G look like is really necessary. The PDE's come from this Lagrangian, in case it's easier to go straight from the Lagrangian to some constant of motion. I've tried the Hamiltonian as first integral but that doesn't seem to work.

$$L = \sqrt{ \frac{H}{G} + {U'}^2 }$$

P.S.: I'm a first time poster so sorry if this is the wrong place to post or any other mistake I made. Also I'd imagine this is a though place to start with the problem, so any help how PDE's are usually solved with first integrals is welcome too - I looked at some threads already but the set of PDE's seems to be a lot simpler in those threads.

Extra Info: I'm doing all the calculations in chapter 2 of the paper (on Wilson loops/Anti-de Sitter Supergravity): http://arxiv.org/pdf/hep-th/9803135v2.pdf

2. Jul 2, 2014