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The problem is simple to solve analytically. Here we use Mathematica to obtain numerical results.

We study various physical properties of a diver who jumps and falls freely from an airplane at a moderate altitude before pulling the ripcord of a parachute.

The constancy of the period of a pendulum, when the amplitude of the oscillations is small, is said to have been discovered by Galileo Galilei (1564–1642) circa 1583 by comparing the period of the oscillations of a swinging lamp in a Pisa cathedral with his pulse rate. This property led Galileo and the Dutch mathematician, astronomer, and physicist Christiaan Huygens (1629–1695) to use a pendulum as a clock regulator.

The differential equation $$ \frac{{d^2 x}} {{dt^2 }} + \lambda (x^2 - 1)\frac{{dx}} {{dt}} + x = 0 $$ , describes the dynamics of the first relaxation oscillator named after the Dutch electrical engineer Balthasar van der Pol (1889–1959) [59, 60]. It is a harmonic oscillator that includes a nonlinear friction term λ ( x ^2 − 1) ẋ . If the amplitude of the oscillations is large, the amplitude-dependent “coefficient” of friction λ x ^2 − 1), is positive, and the oscillations are damped. As a result, the amplitude of the oscillations decreases, and the amplitude-dependent “coefficient” of friction eventually becomes negative, corresponding to a sort of antidamping. If we put x _1= x and x _2=ẋ, the van der Pol equation takes the form $$ \frac{{dx_1 }} {{dt}} = x_{2,} \frac{{dx_2 }} {{dt}} = - x_1 - \lambda (x_1^2 - 1)x_2 $$

In 1873 the Dutch physicist Johannes Diederik van der Waals (1837–1923) obtained his doctoral degree for a thesis on the continuity of the gas and liquid state in which he put forward his famous equation of state that included both the gaseous and liquid states. He showed that these two states could merge in a continuous manner and are in fact the same, their only difference being of a quantitative and not of a qualitative nature. These results were considered very important,^1 and he was awarded the Nobel Prize in Physics in 1910 “for his work on the equation of state for gases and liquids.”

Animal groups display a variety of remarkable coordinated behaviors. For example, all the members in a school of fish change direction simultaneously without any obvious cue; in the same way, while foraging, birds in a flock alternate feeding and scanning. Self-organized motion in schools of fish or flocks of birds is not specific to animal groups. Pedestrian crowds display self-organized spatiotemporal patterns that are not imposed by any controller: on a crowded sidewalk, pedestrians walking in opposite directions tend to form lanes along which walkers move in the same direction.