# nonlinear physics

## Nonlinear physics

The study of situations where, in a general sense, cause and effect are not proportional to each other; in other words, if the measure of what is considered to be the cause is doubled, the measure of its effect is not simply twice as large. Many examples have been known in physics for a long time, and they seemed well understood. Over the last few decades, however, physicists have noticed that this lack of proportionality in some of the basic laws of physics often leads to unexpected complications, if not to outright contradictions. Thus, the term nonlinear physics refers more narrowly to these developments in the understanding of physical reality.

#### Linearity in nonlinear systems

When a large number of particles starts out in a condition of stable equilibrium, the result of small external forces is well-coordinated vibrations of the whole collection, for example, the vibrations of a violin string, or of the electric current in an antenna. Each collective motion acts like an independent oscillator, each with its own frequency. In more complicated systems, many vibrational modes can be active simultaneously without mutual interference. A large dynamical system is, therefore, described in terms of its significant degrees of freedom, thought to be only loosely coupled. The motion of any part of the whole becomes multiperiodic; for example, a water molecule has bending and stretching vibrations with different frequencies, and both are coupled with the rotational motion of the whole molecule. *See* Degree of freedom (mechanics), Molecular structure and spectra, Vibration

#### Failure of perturbation theory

H. Poincaré discovered at the end of the nineteenth century that for many problems this perturbation theory is not entirely satisfactory. He showed, in the case of the Moon's motion around the Earth, that the disturbance by the Sun is strong enough that this standard mathematical procedure fails. The main culprits are resonances, which occur when the frequencies of different degrees of freedom are combined through their nonlinear coupling. A key nonperturbative phenomenon is known to engineers as phase lock: When different frequencies arise in simple multiples of one another, the whole dynamical system falls into a dynamical trap; and for a continuous range of initial conditions, the interaction changes the frequencies of the individual degrees of freedom sufficiently to “lock” the motion into the resonance. *See* Resonance (acoustics and mechanics)

#### KAM theorem

In the 1950s, A. N. Kolmogoroff provided a first account of how the addition of a weak coupling generates chaotic regions and islands in phase space. This problem was later worked out in detail by V. Arnold and J. Moser to yield the KAM theorem. This theorem gives detailed information about the loss of the regular structure as the strength of the coupling increases. It does not say anything, however, about the trajectories in the newly created areas of chaotic behavior. These further investigations are the main goal of such fields as chaos or complexity. The impact of Poincaré's general arguments and the KAM theorem reaches into every area of nonlinear physics. The oldest among the areas is hydrodynamics, where the phenomenon of turbulent flow has so far resisted any effective control. This is what makes weather prediction so difficult. Signal propagation along the nerves and transmission of pulses through synaptic connections are other well-known nonlinear processes. *See* Chaos, Nonlinear acoustics, Nonlinear optics