A phase diagram (or phase portrait) for simple harmonic motion plots velocity \(v\) versus displacement \(x\).
Instead of showing how \(x\) changes with time on a standard \(x\)-vs-\(t\) plot, phase space shows the system’s state as a single point
\((x,v)\). As time evolves, the point traces a curve, revealing conserved quantities and stability.
Undamped SHM. For an ideal oscillator,
\[
x(t)=A\cos(\omega t+\phi),\qquad v(t)=\frac{dx}{dt}=-A\omega\sin(\omega t+\phi).
\]
If we normalize these expressions,
\[
\frac{x}{A}=\cos(\omega t+\phi),\qquad \frac{v}{A\omega}=-\sin(\omega t+\phi),
\]
and square and add, we get
\[
\left(\frac{x}{A}\right)^2+\left(\frac{v}{A\omega}\right)^2
=\cos^2(\omega t+\phi)+\sin^2(\omega t+\phi)=1.
\]
Therefore the phase trajectory is an ellipse:
\[
\frac{x^2}{A^2}+\frac{v^2}{(A\omega)^2}=1.
\]
When \(A\omega=A\) (i.e., \(\omega=1\) in chosen units), the ellipse is a circle. The motion is periodic, so the point goes around the ellipse
forever without spiraling inward or outward.
Energy interpretation. For a mass–spring oscillator with spring constant \(k\) and mass \(m\),
\[
E = \tfrac12 kx^2 + \tfrac12 mv^2
\]
is constant in the undamped model. Level sets of \(E\) in \((x,v)\) space are ellipses, which matches the result above.
Bigger amplitudes correspond to higher total energy and larger ellipses.
Damped motion. When damping is present, mechanical energy decreases over time and the phase point spirals toward the origin
\((0,0)\), the stable equilibrium. A common decaying-amplitude model is
\[
x(t)=A e^{-\gamma t}\cos(\omega t+\phi),
\]
and differentiating gives
\[
v(t)=-A e^{-\gamma t}\Big(\gamma\cos(\omega t+\phi)+\omega\sin(\omega t+\phi)\Big).
\]
The factor \(e^{-\gamma t}\) shrinks both \(x\) and \(v\) as \(t\) increases, so the orbit contracts, producing an inward spiral.
Larger \(\gamma\) means faster contraction.
Phase diagrams are widely used to classify dynamical systems: closed orbits indicate conservative periodic motion, spirals indicate damping or growth,
and fixed points indicate equilibria. University-level extensions include nonlinear oscillators, driven systems, and 2D/3D phase portraits.
