Calculus×Computing
ES

Second-order linear equations (oscillations)

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

mx¨+cx˙+kx=F(t)m\ddot x + c\dot x + kx = F(t): springs, pendulums, circuits, suspensions. The roots of the characteristic equation mr2+cr+k=0m r^2 + c r + k = 0 decide whether the system oscillates (complex roots), returns smoothly (real roots) or resonates.

Formulas

x¨+2ζω0x˙+ω02x=0,ω0=k/m,ζ=c2km\ddot x + 2\zeta\omega_0\dot x + \omega_0^2 x = 0, \qquad \omega_0 = \sqrt{k/m}, \quad \zeta = \frac{c}{2\sqrt{km}}
x(t)=Ae−ζω0tcos⁡(ωdt+φ),ωd=ω01−ζ2  (ζ<1)x(t) = A e^{-\zeta\omega_0 t}\cos\big(\omega_d t + \varphi\big), \quad \omega_d = \omega_0\sqrt{1 - \zeta^2}\ \ (\zeta < 1)
underdamped response

Where it shows up in computing

  • Classical mechanics★★★★★fundamentalPhysics and simulation

    The harmonic oscillator is the model of every small vibration around an equilibrium.

  • PID control★★★★★frequentRobotics and control

    Tuning a controller is choosing the damping ratio ζ\zeta of the closed loop: fast without overshoot.

  • Physics engines★★★★★frequentPhysics and simulation

    Springs and dampers drive cloth, ragdolls, camera smoothing and vehicle suspensions.

Where it shows up in AI

  • Momentum and Adam★★★★★advancedAI and machine learning

    Gradient descent with momentum is a discretized damped oscillator (heavy-ball ODE) rolling in the loss landscape.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

⚙ Robotics and control

ℒ AI and machine learning

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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