Finding the first integral for equations of motion

In summary, the person is seeking help finding the first integral of motion for a set of PDEs they are not familiar with. The equations involved are quite complicated, but the paper they are working from says the calculation is straightforward. The person provides equations for H and G, and mentions a solution that involves a constant. They also mention trying to use the Hamiltonian as a first integral, but it did not work. They are looking for help with solving PDEs using first integrals.
  • #1
Elodin
1
0
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)


[tex]
\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}
[/tex]

[tex]
\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}:
[/tex]

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

[tex]
\mbox{I know the solution is: }
\begin{equation}
\left(\frac{G}{H} \right)^2
U^{'2}
+ {G \over H} = \mbox{constant}.
\end{equation}
[/tex]

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.

[tex]\begin{equation}
L = \sqrt{ \frac{H}{G}
+ {U'}^2 }
\end{equation}[/tex]

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
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 
  • #3
Yes, I also want to some additional information here. I'm waiting for it.
 
1.

What is the purpose of finding the first integral for equations of motion?

The first integral for equations of motion is a mathematical tool used to simplify and solve complex systems of differential equations. It allows us to identify conserved quantities, such as energy or momentum, in the system and provides insight into the behavior of the system over time.

2.

How is the first integral related to the Lagrangian and Hamiltonian formulations of mechanics?

The first integral is closely related to both the Lagrangian and Hamiltonian formulations of mechanics. In the Lagrangian formulation, the first integral is the conserved quantity that corresponds to the system's symmetry, while in the Hamiltonian formulation, it is the constant of motion associated with the system's Hamiltonian function.

3.

What are the steps for finding the first integral for a system of equations of motion?

The first step is to identify the conserved quantity that corresponds to the system's symmetry. Then, we use the Lagrange or Hamilton equations to express the conserved quantity as a function of the system's variables. Finally, we solve for the first integral by integrating the resulting expression.

4.

Can the first integral be used to predict the behavior of a system over time?

Yes, the first integral can provide valuable information about the behavior of a system over time. It can help us identify stable or periodic solutions and provide insight into the overall dynamics of the system.

5.

Are there any limitations to using the first integral for equations of motion?

While the first integral is a powerful tool, there are some limitations to its use. It is only applicable to systems with a certain degree of symmetry, and it may not always be possible to find a first integral for a given system. Additionally, the first integral may not provide a complete understanding of the system's behavior and may need to be combined with other methods for a more comprehensive analysis.

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