I'm doing some exercices to see how are related the hamiltonian and the gradient systems. I did an exercise, but I don't know if my approach is correct and I have a few questions about it. Let's state it:
Let $H:\mathbb{R}^2 \to \mathbb{R}$ be a $C^2$ function. We consider the Hamiltonian:
\begin{cases}\dot{x}=-\frac{\partial H(x,y)}{\partial y}\\ \dot{y}=\frac{\partial H(x,y)}{\partial x}\end{cases}
and the gradient systems:
\begin{cases}\dot{x}=\frac{\partial H(x,y)}{\partial x}\\ \dot{y}=\frac{\partial H(x,y)}{\partial y}\end{cases}
a) Given an ODE on the plane, we'd say that a simple critical point is a critical point whose eigenvalues are different of $0$. Prove that if we have a simple critical point for one of those systems, it's also a simple critical point for the other one.
So here's my try:
Let's suppose that we have a simple critical point for a hamiltonian system. We know that if $(x,y)$ is a simple critical point, we have
\begin{cases}\dot{x}=-\frac{\partial H(x,y)}{\partial y}=0\\ \dot{y}=\frac{\partial H(x,y)}{\partial x}=0\end{cases}
It's straightforward that:
\begin{cases}\frac{\partial H(x,y)}{\partial x}=0\\ \frac{\partial H(x,y)}{\partial y}=0\end{cases}
so it's a simple critical point of the gradient system too.
It's correct?
The condition of "simple" means that $(x,y)\neq (0,0)$?
Thanks.
Given a system \begin{cases} \dot x = F^1(x, y),\\ \dot y=F^2(x, y) \end{cases} and a critical point $(x_0, y_0)$, we consider the Jacobian matrix
$$ DF(x_0, y_0) = \begin{pmatrix} F^1_x(x_0,y_0) & F^2_x(x_0,y_0) \\ F^1_y(x_0,y_0) & F^2_y(x_0,y_0) \end{pmatrix}$$
the critical point is called simple, if the eigenvalue of this matrix are both nonzero. Note that this is the same as saying that $DF$ has nonzero determinant.
Now you are given both the Hamiltonian system $(Ha)$ and the gradient system $(G)$ of a function $H$. You checked already that $(x_0, y_0)$ is critical point of $(Ha)$ iff it is a critical point of $(G)$.
Now try to calculate the Jacobian matrix of both system. For the gradient system, you would get the Hessian matrix of $H$. For the Hamiltonian system, you got a very similar matrix. You can check that they have the same determinant.