Mathematically, this is called sensitivity to initial conditions. A set of initial values of variables that is slightly different from another set can result in completely different outcomes in a chaotic system. The implication is that unless we can measure the conditions of a complex system exactlyat one time, we cannot predict its detailed behavior at a much later time even if the system is governed exactly by a known set of physical principles. This applies to the orbits of asteroids and moons in our solar system, and it should apply as well to the behavior of people even if they have no free will. While the behavior of a system may be predictable over a certain time period whose duration depends on the complexity of the system, all but the simplest systems are eventually subject to chaos. But can we, at least in principle, measure all the positions, velocities, and forces exactly? The answer is “no.” On very tiny size scales, Heisenberg’s uncertainty principle applies: the position and velocity of a particle – a tiny piece of matter – cannot both be known precisely (see Ch. 7). Therefore, while it is possible that we are machines without true free will, we at least are not completely predictable machines!
Chapter 4: Motions According to Newton 4-14SummaryThe Enlightenment period saw the development of a new theory of motion by Kepler, Galileo, and Newton. Newton’s three laws of motion and Law of Universal Gravitation allowed the understanding of the orbital laws discovered by Kepler. They also provided the formulation by which future positions and velocities of objects could be predicted given their masses as well as their positions and velocities at a specific time. This is valid for motions in space as well as on the Earth. The same equations are still in wide use, although we now know that they are inaccurate under extreme conditions, such as near extremely compact, very massive objects or when the speed approaches that of light. In cases in which the motion is complex, such as two-dimensional trajectories, rational analysis – in which the problem is separated into sub-problems that are each solved and then combined into an overall solution – is often very useful. This approach is a very important tool of the modern scientist.An important concept introduced in this chapter is conservation, which means that the value of certain key quantities does not change unless some external force acts on the system. Two conserved quantities are momentum– the product of mass and velocity – and energy. Energy has various aspects; two of these are kinetic energycorresponding to motions – which includes the heat energy of microscopic motions – and stored potential energy. The energy added or subtracted by an external force is called work. Energy can be converted from one form to another. An application of conservation of momentum is the acceleration of a rocket forward while the exhaust is propelled backward. Examples of conservation of energy include the increase in speed of a falling object and the back-and-forth motion of a pendulum.