Most people try to optimize by working harder. More effort, more results. But the physics of driven systems suggests a different priority: phase alignment and low dissipation dominate raw effort. To see why, consider the simplest case.

The Setup

A driven damped oscillator has three knobs:

  • $F_0$ – the driving force amplitude, i.e., how hard you push (N)
  • $\phi$ – the phase angle between the driving force and the system’s motion (rad)
  • $\gamma$ – the damping co-efficient, i.e., how much energy the system dissipates (s⁻¹)

The steady-state energy stored in the system is given by:

\[E = \frac{F_0^2 \cos^2(\phi)}{8\gamma^2 m \omega_0^2}\]

If you analyze that formula carefully, it tells you everything!

The Hierarchy

Phase alignment ($\phi$) is a gate. $\cos^2(\phi)$ ranges from 1 to 0. Push in sync and you get full energy transfer. Push at $\phi = \frac{\pi}{2}$ and you transfer nothing, regardless of force magnitude.

Damping ($\gamma$) is an amplifier. Energy scales as $\frac{1}{\gamma^2}$. Halve the dissipation and you store four times the energy. As damping approaches zero, stored energy diverges toward infinity. This is why resonance disasters happen: bridges collapse, wine glasses shatter. Tiny forces, negligible dissipation, and pushes that remain closely in phase.

Driving amplitude ($F_0$) is quadratic leverage. Energy scales as $F_0^2$. It matters, but it can’t compensate for poor phase alignment. At $\phi = \frac{\pi}{2}$, no amount of force transfers energy.

The Takeaway

The formula confirms what’s counterintuitive: effort ($F_0$) is quadratic, but phase ($\phi$) is the only parameter that can zero out transfer completely, and low damping ($\gamma$) compounds small inputs into large stored energy.

The analogy extends beyond oscillators. Most people try to optimize by working harder: more amplitude. However, the highest-leverage moves are finding systems where your effort is naturally in phase with how the system wants to move, and reducing the dissipation that bleeds your invested energy away. Then you turn up the effort.