I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an example of a non-conservative system, which I've attempted to provide in the following way.
So, given that \begin{equation}\ddot{\textbf{x}}=\textbf{f}(\textbf{x})|\textbf{x}\in E^2 \end{equation} and \begin{equation}\ddot{\textbf{x}}=-\frac{\partial U}{\partial \textbf{x}} \end{equation} Is the function \begin{equation}\textbf{f}(\textbf{x})=\dot{\textbf{x}} \end{equation} an equation describing a non-conservative system?
Please provide an explanation as well.