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:

• g=0, the trajectories are a family of curves closed around the equilibrium point q =dq /dt=0.
• g¹ 0, the trajectories end at (g>0) or leave (g<0) the equilibrium point.

We can demonstrate that these trajectories are spirals for ½ g½ << w.