Math 6410 • ODE

\textbf{Linearization.}
Assume \(g\in C^\infty(\R^n)\) has a strict local minimum at the origin. Consider the
second-order system
\[
  \ddot{x}+B\dot{x}+\nabla g(x)=0,
\]
where \(x(t)\in\R^n\) and \(B\in\R^{n\times n}\) is a symmetric positive definite matrix.
\begin{enumerate}
\item[(a)] Rewrite the system as a first-order system.
\item[(b)] Write the linearization of the system at the origin.
\item[(c)] Prove that the origin is an asymptotically stable fixed point.
\end{enumerate}