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The three fundamental equations of SHM follow from successive differentiation: $x = A\sin(\omega t + \phi)$, $v = A\omega\cos(\omega t + \phi)$, and $a = -A\omega^2\sin(\omega t + \phi)$. Velocity leads displacement by $\pi/2$ and acceleration leads by $\pi$. At the mean position ($x = 0$), velocity is maximum ($A\omega$) and acceleration is zero. At the extremes ($x = \pm A$), velocity is zero and acceleration is maximum ($A\omega^2$).

The velocity-displacement relation $v = \omega\sqrt{A^2 - x^2}$ is often the fastest route to solving problems, bypassing the need to find phase. The acceleration-displacement graph is a straight line through the origin with slope $-\omega^2$ — the definitive graphical test for SHM. The velocity-displacement graph forms an ellipse with semi-axes $A$ and $A\omega$. From $v_{\max} = A\omega$ and $a_{\max} = A\omega^2$, we can extract $\omega = a_{\max}/v_{\max}$, $A = v_{\max}^2/a_{\max}$, and $T = 2\pi v_{\max}/a_{\max}$. These relations are frequently tested when problems provide maximum velocity and acceleration instead of direct SHM parameters.