Elements of Classical Mechanics

So confident were the researchers that Hamel would write in the 1920s in the famous Handbuch der Physik} an axiomatic treatment of mechanics—an axiom is a statement that is generally accepted as self-evident troth. I follow Hamel’s lead

and delineate the basic elements of classical mechanics:

1) Material body: A body is a three-dimensional, differentiable manifold whose elements are called particles. It possesses a nonnegative scalar measure that is called the mass distribution of the body. In particular, a body is called rigid if the distances between every pair of its particles are time invariant.

2) Force: The force describes the action of the outside world on a body and the interactions between different parts of the body. We distinguish between volume forces and surface forces.

3) Euclidean space-time: The interaction of the forces with the material body occurs in space and time and is called an event. Events in classical mechanics occur in Euclidean space-time. The Euclidean space exhibits a metric that abides, for infinitesimal displacements d. s, the law of Pythagoras over the three-dimensional space {xі, X2, хз):

з

dsz = dx + dxj + dx — ^ dxf

i=l

The concept of a particle, so important in classical mechanics, defines a math­ematical point with volume and mass attached to it. We could also call it an atom or molecule, but prefer the mathematical notion to the physical meaning. By accu­mulating particles we form material bodies with volume and mass. If the particles do not move relative to each other, we have the all-important concept of a rigid body.

Without forces, the body would, according to Newton’s first law, persist at rest or continue its rectilinear motion. However, we shall have plenty of opportunity to model forces. There are aerodynamic and propulsive forces acting on the out­side of the body as surface forces. We will deal with gravitational effects, which belong to the volume forces, acting on all particles, and not only on those at the surface.

In classical mechanics space and time are entirely different entities. Space has three dimensions with positive and negative extensions, but time is a uniformly increasing measure. For us, this so-called Galilean space-time model will suffice. However, we should remember that in 1905, just after the turn of the century, Albert Einstein revitalized physics with his Special Theory of Relativity, where time becomes just a fourth dimension.

Einstein did not abolish Newton’s laws, but expanded the knowledge of space and time. He relegated Newton to a sphere where velocities are much less than the speed of light. However, that sphere encompasses all motions on and near the Earth. Even planetary travel is adequately represented by Newtonian dynamics, consigning relativistic effects to small perturbations.