Without friction a simple linear equation expresses the amount of energy you need to accelerate a hockey puck. With friction the relationship gets complicated, because the amount of energy changes depending on how fast the puck is already moving.
Nonlinearity means that the act of playing the game has a way of changing the rules. You cannot assign a constant importance to
friction, because its importance depends on speed. Speed, in turn, depends on friction. That twisted changeability makes nonlinearity hard to calculate, but it also creates rich kinds of behavior
that never occur in linear systems. In fluid dynamics, everything
boils down to one canonical equation, the Navier-Stokes equation.
It is a miracle of brevity, relating a fluid's velocity, pressure, density, and viscosity, but it happens to be nonlinear. So the nature
of those relationships often becomes impossible to pin down. Analyzing the behavior of a nonlinear equation like the Navier-Stokes equation is like walking through a maze whose walls rearrange
themselves with each step you take. As Von Neumann himself
put it: "The character of the equation ... changes simultaneously
in all relevant respects: Both order and degree change. Hence, bad
mathematical difficulties must be expected." The world would be
a different place—and science would not need chaos—if only the
Navier-Stokes equation did not contain the demon of nonlinearity.
— James Gleick
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