For example, if a pendulum system in the real world oscillates its amplitude and period will eventually decrease. Does that still count as a periodic motion? Since the motion doesn't have the same amplitude at equal time periods.


2 Answers 2


The motion of pendulum described in the OP is often referred to as "damped oscillations". In this sense it is fair to say that "oscillatory" is used in broader sense than "periodic", which implies exact repetition.

In response to a similar question I would like to add some clarifications, scattered over the comments:

Periodic motion repeats itself after certain time, called period, $T$: $$ x(t+T) = x(t) $$ A motion of a pendulum (without damping) or a rotation of a planet around a star are periodic motions.

Oscillatory motion usually means that the periodic motion retracing the same trajectory, such as that of a pendulum. One would however often apply the term to motions that are not strictly periodic: e.g., one often talks of damped oscillations, meaning that the motion would be oscillatory without damping.

Harmonic implies linear restoring force, which leads to solutions in terms of harmonic functions, such as sines and cosines. Thus, $$ F=-kx,\\ m\ddot{x} + kx=0 $$ is a harmonic motion. On the other hand $$ F=-kx^3,\\ m\ddot{x} + kx^3=0 $$ is not harmonic - it will be often described as a non-linear oscilator, as opposed to linear=harmonic motion.

Finally, $$ m\ddot{x} +\gamma \dot{x} + kx^3=0 $$ is an example of a damped non-linear oscillator. The motion described by this equation is not periodic, and in some cases is not even quasi-periodic (overdamped oscillator), but the analogy with the oscillator equation is obvious.

  • $\begingroup$ Does that mean the motion of pendulum in the real world is not periodic but we tend to say that all oscillatory motions are periodic in a broader sense? $\endgroup$ Apr 26, 2021 at 12:57
  • $\begingroup$ They are approximately periodic. One talks about damped oscillations and even overdamped oscillations, when there is no periodicity at all! It is really an abuse of language, but the idea is that these are actually related to a periodic motion that we would have without damping. $\endgroup$
    – Roger V.
    Apr 26, 2021 at 13:10
  • $\begingroup$ So...that means that if these oscillations don't experience damping, they are periodic motions and that's why we also consider them periodic motions?Sorry, but i'm just really confuse at this point. $\endgroup$ Apr 26, 2021 at 13:19
  • $\begingroup$ Oscillations is just not a rigorously defined term. It literally means periodic motion, but, as I explained, it is often applied to the situations where it is not exactly the case. $\endgroup$
    – Roger V.
    Apr 26, 2021 at 13:21
  • $\begingroup$ Ah i think i get it now but just to make sure this means that oscillatory motions are periodic motions but there are some cases where the motion is not periodic but we still call it an oscillatory motion? $\endgroup$ Apr 26, 2021 at 13:36

As pointed out in a comment in another answer to this question: 'oscillation' is not a rigorously defined term. But there is enough convention that the word is a very usable word.

From a physics point of view:
There are two conditions that are necessary in order to have oscillation at all.
-There must be a form of restoring effect present, such that there is a lowest state, and when the system is not in that lowest state the restoring effect will tend to force it back to that lowest state.
-There must be a form of motion, such that if some part of the system has acquired a velocity then a force is required to change that velocity. (Velocity will make the moving part of the system overshoot the lowest point, sustaining the oscillation)

Let's start with the simplest case: only one degree of freedom.

Example: a weight attached to a coiled spring, the spring alternates between being elongated and being compressed.

In the idealized case the force exerted by the spring is according to Hooke's law, and then the oscillation is harmonic oscillation.

Realistically the force exerted by the spring does not change linear with displacement. I think that in that case the oscillation will still be periodic. It won't be quite harmonic oscillation, but after a full cycle everything is back to were it was a cycle ago, so the same should happen again.

So: the two conditions that are necessary in order to have oscillation at all are precisely conditions that tend to make the resulting motion periodic.

I think introducing a second degree of freedom is when we see opportunity for non-periodic behavior.

If memory serves me: a tabletop gadget exists that is a spherical pendulum with a repelling magnet underneath the lowest point of the pendulum bob.

You pull the pendulum bob away from the center and you release it. Magnetic force is very short range; the motion of the pendulum bob is affected only close to the center. The result is chaotic looking motion. The pendulum swings, but the motion is not periodic.

Another instance of the theme 'adding a degree of freedom' is the difference between the two-body-problem and the three-body-problem in the case of gravity.

The two-body-problem is the Kepler problem, the solution is periodic orbital motion. The general three-body-problem does not have solutions that are long term periodic.


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