# How to check if ode system is gradient system?

How do I check for a given (nonlinear) system of ODEs if this is a gradient system? Meaning how do I check the existence of a pseudo Riemann metric and a potential function? May be somebody could post a simple example or give a literature hint?

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What do you mean by gradient system? – Mercy King Jun 25 '12 at 19:24
In the easiest case: planetmath.org/GradientSystem.html more general: books.google.de/… – litro Jun 26 '12 at 11:41
$F=(F_1,\ldots,F_n) \in C^1(\mathbb{R}^n,\mathbb{R}^n)$ has a potnetial means that there is a function $V \in C^1(\mathbb{R}^n,\mathbb{R})$ such that $F(x)=\nabla V(x)$ for every $x \in \mathbb{R}^n$. Since $\mathbb{R}^n$ is simply connected, there exists such a $V$ iff $\partial_iF_j=\partial_jF_i$ for every $i<j$. As an example, take $F(x,y)=(y,x)$. Clearly $\partial_1F_2=\partial_2F_1$, therefore $F$ has a potential. In fact $V(x,y)=xy$ is a potnetial.If you replace $\mathbb{R}^n$ by $\mathbb{R}^n\setminus 0$ it's a different story though! – Mercy King Jun 26 '12 at 12:51
with \delta_i F_j being the derivative with respect to the i-th variable of the partial derivative with respect to the j-th ? – litro Jun 26 '12 at 14:07
$\partial_iF_j=\partial F_j/\partial x_i$ – Mercy King Jun 26 '12 at 17:09