The following problem occurred on a homework assignment last week, and I wanted to know how to do the analysis to prove it in the nontrivial case.
Consider a two dimensional system with equillibrium point $0$ and smooth conserved quantity $E : \mathbb{R}^2 → \mathbb{R}$. Suppose that $DE(p) \neq 0$ for all $0 < |p| < 1$. Prove that $0$ is stable if and only if E has a strict max or min at $0$.
We say $y' = \phi(y)$ (where $\phi: \mathbb{R}^d\rightarrow\mathbb{R}^d$) is a conservative dynamical system if there is a continuously differentiable $E:\mathbb{R}^d\rightarrow\mathbb{R}^d$ such that $E\circ f$ is constant for any solution $f$ of the system and $E$ is not constant on any nontrivial open set.
My answer on the homework was "Let $\phi \equiv 0,$ then the claim is false." So my questions are as follows:
- If we say $\phi$ is not identically $0$, is the claim now true?
- If so, how do we prove it? I've proven that $E$ having a strict max/min leads to stability, but I'm stuck on the analysis for the converse. My attempt is as follows:
Let $\psi:\mathbb{R}\times\mathbb{R}^2\rightarrow\mathbb{R}^2$ be the flow of the system. Suppose that $E(0,0) = E_0$ is stable. Then for any neighborhood $N_\varepsilon$ of $(0,0),$ there is a neighborhood $N_\delta \subset N_\varepsilon$ of $(0,0)$ such that if $q \in N_\delta$ then $\psi(t,q) \in N_\varepsilon$ for all $t \in \mathbb{R}.$ But since conservative systems have no asymptotically stable points (theorem proven in class), we have that $q$ is periodic. Since $E$ is a conserved quantity, $\psi(t,q)$ is a level curve of E, and since it is periodic, it must be a simple closed curve. So close to $(0,0),$ we can generate a contour plot of level curves which close in on a point; given that the gradient of $E$ is not zero on some punctured neighborhood of $(0,0),$ it must be the case that $E(0,0)$ is either a strict maximum or strict minimum. In particular, $(0,0)$ is not a saddle point of $E.$
My dissatisfaction with the above proof is that I haven't been able to come up with a $\delta-\epsilon$ style argument that the contour plot in the neighborhood of $0$ implies a strict maximum/minimum of $E.$