# For a motion to be simple harmonic motion, must it be periodic?

Is it necessary for a motion to execute simple harmonic motion to be periodic? Can't simple harmonic be non-periodic?

• Is your question related to a forced harmonic oscillator, which could have an aperiodic source function that drives the motion? Jun 9, 2020 at 10:35
• @Ishika-96-sparkle No. I wanted to know that in real life like a pendulum which does not have constant energy supply its motion eventually stops and with time periods its amplitude also decreses so it means that its motion is not periodic . So i wanted to know that the motion of that pedulum will be Simple harmonic motion or not as it is non- periodic Jun 9, 2020 at 13:21
• Amplitude dying out over time would not be an indication of periodicity but the wavelength of oscillation is. If the wavelength cannot be ascribed over which a function repeats itself, then its can be said to non-periodic. A pendulum dying out on its own is periodic with diminishing amplitude. Jun 9, 2020 at 13:33
• @Ishika_96_sparkle it means then for a motion to be periodic it should have same frequency and wavelength over time periods. A periodic motion can have decreasing or changing amplitude over time periods.For a motion to be periodic conditions are same frequency and wavelength not amplitude. Can you please correct me if I am wrong? I will be very thankful to you because I am really struggling with the concept and conditions of periodic motion Jun 9, 2020 at 13:37
• Mathematically speaking, a periodic function is given by $f(x+T)=f(x)$ i.e. if the argument is changed by an amount T then the we get the same function again .e.g. $\sin(x+2\pi)=\sin(x)$. This is called transnational symmetry of the function. So, $2\pi$ is the wavelength of one full circulation of the angle after which it repeats the same values. Jun 9, 2020 at 13:43

Simple harmonic motion is necessarily periodic since its is described by the function $$x(t)=A\cos(\omega t+\varphi)\, .$$

The converse is not true: periodic motion is not necessarily simple harmonic motion. There are multiple examples of this.

Planetary motion is periodic but not a simple harmonic motion: it is 2D and takes place on an ellipse. For instance: \begin{align} x(t)=A\cos(\omega t)\, ,\qquad y(t)=A\cos(\omega’ t) \end{align} will be periodic in $$x$$ but not necessarily form on a closed orbit unless $$\omega$$ and $$\omega’$$ are commensurate.

A combination of simple harmonic motions is also periodic if the two motions have commensurate frequencies, but the sum is not a simple harmonic motion. For instance, $$x(t)=A\cos(\omega t)+ B \cos(2\omega t)$$ will be periodic but not simply harmonic. More generally, any signal decomposed using Fourier analysis will be periodic with the period given in terms of the fundamental frequency (v.g. musical notes) but not simply harmonic.

Is it necessary for a motion to execute simple harmonic motion to be periodic?

Simple harmonic motion is motion that can be written as a sinusoidal function. Sinusoidal functions are periodic. So yes it is necessary

Every motion that repeats itself is essentially periodic. However, that a motion is periodic does not mean it is also simple harmonic. Simple harmonic motion is any periodic motion where the restoring force is always proportional to the displacement of the body from equilibrium. The above condition means we can describe the motion completely- the position, velocity, acceleration, period and frequency. If any periodic motion satisfies the condition $$F=-k\,x$$ or approximates to the above given some reasonable assumptions, then we call it SHM. However, there is almost nothing we can say about periodic motion if it is not SHM except of course that it repeats. The bottom line is: while all SHM is periodic, not all periodic motion is SHM.