copyright 2017 Jennifer Coopersmith
Newton’s attraction or repulsion of point masses, in accordance with his Second Law of Motion, was one of the most momentous advances in physics – explaining such a wealth of phenomena (apples falling, the Moon orbiting the Earth, billiard ball collisions, and more) in just one way (a mass accelerating along the direction of the force). It might all have been so much more complicated: for example, instead of just accelerating, the mass might have swollen; or there might have been three-body forces; or the mass might have moved in proportion to the rate of change of the rate of change of velocity – and so on.
But does this beautiful simplicity, of reducing the system to interacting pairs of masses, really cover all scenarios? Already, when we come to extended rotating bodies, even while they are assumed to be rigid, we must apply different formulae (for the ‘moment of inertia’) depending on the exact shape of the body. Making things even more realistic – non-uniform densities, and bodies that can squash, stretch, twist or bend – we need to consider not just the mass distribution but the ‘energy density’ on a point by point basis. When, furthermore, materials can be electrically charged, or magnetic, or exposed to an external field, then things get more complicated again. (In fact, there is a family connection between engineering, materials science, and Einstein’s General Relativity; and engineers have been using techniques such as the Principle of Virtual Work years ahead of most physicists.)
In these more realistic cases the simple picture of ‘point masses’ (particles) and ‘forces’ does not suffice (although, in most physics courses, the extent to which problems in Newtonian Mechanics have been cherry-picked for their tractability is not usually mentioned). It is not simply a question of scaling up from a handful of particles to, say, a trillion trillion particles. No – in statistical mechanics, physical chemistry, engineering, quantum mechanics, Gravitation, and so on, the whole system must be explained in one go. The various parts act together in concert, and not simply as a summation over particle pairs.
In this whole-system approach, the new elements are more complicated and less intuitive than point masses and forces – but it cannot be helped. The new elements are the various kinds of energies. In a closed system, the total energy is conserved, and also the difference between the kinetic and potential energy is minimized, at each time, and over the whole time interval of the given problem.