I think I see what you mean. If it's small enough (whatever that is), it will be effectively zero in the real world. In "math modeling world", it never really reaches 0. Like you said, though, if we're really dealing with non-linearities and the imperfectness of the real world, can we ever say it actually reaches zero? Or is this just not measurable and therefore an invalid (and more philosophical) question?Mathematically, the oscillations decrease asymptotically.
But in real life, there are always nonlinearities, so yes it stops.
But also in real life, we have micro earthquakes, so no it doesn't.
It's just a matter of definition. How small do the oscillations get before you call them "zero?"
In the case of a damped pendulum we would typically consider linear damping -- a retarding force that is proportional to the pendulum's velocity. With small (under-damped) linear dampening the motion of the pendulum will turn out to follow a sine wave multiplied by a decreasing exponential.
Is it not a consequence of the fluctuation-dissipation theorem that viscous damping will necessarily be accompanied by fluctuations and that these will cause the motion to reverse (at least for an instant) after a finite time?It all depends upon the kind of damping present. Viscous damping will never totally stop the motion, but dry friction (Coulomb) damping will bring the motion completely to rest.
Of course it does.Does a pendulum with damping effects ever truly stop oscillating?
The graph is part of the model used to describe the behavior of the pendulum. If your model involves a damping factor proportional to the speed then that model will predict an infinite amount of time for the pendulum to stop. By the way, at some point the velocity and the position predicted by the model will need to be known simultaneously with an uncertainty that's less than the Heisenberg Uncertainty. In other words, the model fails outside of its limits of validity.That is, does the graph tend to infinity or actually reach a value of 0, i.e. the equilibrium position?