How would I draw the phase portriat for the following system, and then find the quantities conserved along the flow?
\begin{equation*} \vec{\dot{x}}= \begin{bmatrix}-sin(\theta) & cos(\theta)\\-cos(\theta) & -sin(\theta)\end{bmatrix}\vec{x} \end{equation*}
Firstly I found the eigenvalues to be: $\lambda = -\sin(\theta)+ i\cos(\theta),-\sin(\theta)- i\cos(\theta)$, which would correspond to spirals away/towards the origin depending on the value of $\theta$.
Here is my first question: how do I draw the phase portrait if it is dependent on $\theta$.
and:
How do I find what Quantities are conserved?
Considering the system
$$ \dot X + R X = 0 $$
with $X = (x_1,x_2)^{\top}$ and $R = \left(\begin{array}{cc}a & -b\\ b & a\end{array}\right)$ the solution is given as
$$ X = \left(\begin{array}{cc}\cos(bt) & \sin(bt)\\ -\sin(bt) & \cos(bt)\end{array}\right)\left(\begin{array}{c}C_1\\ C_2\end{array}\right)e^{-at} $$
This represents spirals in the plane $x_1 \times x_2$
Those spirals can be a source $a < 0$ or a sink $a > 0$ or a center $a = 0$
We have also
$$ X^{\top}X = (C_1^2+C_2^2)e^{-2at} $$
or
$$ ||X|| = ||C|| e^{-at} $$