I would like to know how to prove that there is no conserved quantity in a system.
Let $k$ and $c$ be real numbers. Below is the ODE for the damped harmonic oscillator.
$$\ddot{x}=-kx-c\dot{x}$$
From a physics point of view, I would expect that no conserved quantities exist in this system, since energy is not conserved in it.
The definition of a conservative quantity is a non-constant function $f:\mathbb{R}^2\to\mathbb{R}$ that satisfies $\frac{\mathrm{d}f(x(t),y(t))}{\mathrm{d}t}=0$ for all solutions $(x,y)$ of the following system.
$$
\begin{aligned}
\dot{x}&=y\\
\dot{y}&=-kx-cy
\end{aligned}$$
Best Answer
Hint: If you manage to show that all flow-lines converge to the same point $(x_0,y_0)$ then any continuous function which is conserved along flow-lines must take the same value, i.e. the value at $(x_0,y_0)$.