Something bordering on the mysterious

In Wigner's view, mathematics is a “wonderful gift that we neither understand nor deserve.”

In a famous essay, Eugene Wigner once speculated about the “unreasonable effectiveness of mathematics in the natural sciences.” Most of mathematics, he argued, finds its inspiration not in direct experience but in the creative impulse of the mathematician, who explores an infinite world of logical possibilities. Yet these 'unnatural' explorations , time and again, prove to be scientifically useful and necessary. In Wigner's view, the amazing value of mathematics is a “wonderful gift that we neither understand nor deserve.”

A related surprise is the way similar mathematical structures so often show up in totally different settings, so that equations originating in physics end up applying in, say, biology. Weirdly enough, for example, the Schrödinger equation (in imaginary time) turns up in the mathematical biology of evolving populations. A less bizarre but equally striking example arises in the curious case of the London Millennium Bridge, which opened on 10 June 2000. A 325-metre steel structure, the bridge was closed three days later so that engineers could investigate a lateral oscillation that emerged whenever more than about 160 people walked on it. As a team of theorists has recently shown (S. Strogatz et al., Nature 438, 43–44; 2005), these vibrations may have a fairly simple origin — one that reveals an unexpected link between seemingly unrelated phenomena.

The bridge has a lateral mode that can be described as a damped harmonic oscillator. People exert small lateral forces when they walk, and so each person on the bridge drives this mode with a weak force proportional to sinθ_i, where θ_i is the phase of the ith person's 'walking cycle'. The force delivered by the full crowd is the sum over all people, and, if they are not marching in step, should be small. But the pedestrians also respond to the motion of the bridge, and this interaction can lead to surprising consequences.

Strogatz et al. suppose that the walking phase θ_i of each person will ordinarily increase at some average rate ω_i (with these ω_i distributed about some mean value). Interaction with the bridge adds an extra term to the time derivative of θ_i, depending on the difference between θ_i and a phase for the oscillating bridge. This leads to an equation that reflects how people, in trying to keep their balance, naturally tend to synchronize their steps with the swaying of the bridge.

Conveniently, the resulting mathematics has been studied extensively elsewhere. It offers, for example, a description of the collective chirping of crickets, who tend to advance or delay their chirping to achieve synchrony with others nearby. In another setting, the same equations describe the dynamics of arrays of coupled Josephson junctions, where now the phase refers to the superconducting wavefunction. Regardless of interpretation, the mathematics works the same, and it reveals the existence of an abrupt phase transition, with random phase behaviour giving way to strong synchronization when the number of oscillators (people, crickets, devices) passes a threshold.

The resulting theory works quite well for the London Millennium Bridge (which was later stabilized with damping devices). It may not settle all questions about how people interact with wobbling bridges, but it's surely another demonstration of the surprising effectiveness of mathematics — in Wigner's words, “something bordering on the mysterious”.