The absorbed oscillator : the pendulum with fluid friction.
Because the pendulum is never ideal, the motion always stop. So it seems obvious to adjust the equation with a term of absorbtion : gdq /dt.
The linearised equation is :
The expression of E is now:
The energy is retained if g=0, decreases if g>0. And because E>0, it decreases to 0.
When the energy is equal to 0, the equilibrium is stable and every motion try to recover this position.
For g<0, dq /dt=q =0 is an unstable equilibrium. When the pendulum moves, the movement is amplified.
The trajectories solution of the equation are tangential to the local speed vector. We can use the variables x=q and y=dq /dt:
We can demonstrate that these trajectories are spirals for ½ g½ << w.