Pedestrian Physics 2

NEWTON, dynamics

Newton’s mechanics is an axiomatic system. As such, all it has to prove is that it is internally consistent. Newton, however, took it one step further: He declared his axioms to be “Laws of Nature”. On this level it is the observer and the experimenter who are the final arbiters of “true” or “false”.

His laws ruled science as absolute truths for more than 200 years. Then, in 1901, Walter Kaufmann found a flaw in Newton’s legal system. Fast and slow electrons, deflected by the magnetic field, behaved as if they obeyed separate laws.

The observation led to a storm of speculations about the shape and nature of electrons, none seemed satisfactory. The answer came from a different direction. Einstein declared that Newton’s definition of mass was incomplete. Newton had defined “mass” as a measure of the “Quantity of Matter” in a body. Einstein found that the “Quantity of Motion” also made a contribution. It was extremely small, but the new definition changed the direction and the face of science.
**
Newton based his system on Euclid’s geometry to which he added the undefined concepts of mass,”m” and time “t”; his three axioms of motion, and his axiom of universal gravitation.

On this basis he created several defined concepts:
In modern notation, the vectors

1/ Velocity: v = ds/dt
2/ Acceleration: a = dv/dt
3/ Momentum: M = mv
4/ Accelerative Force: F = ma
5/ Motive Force: F = dM/dt,

and the scalar:

6/ Work: W = F.s
Newton termed momentum “quantity of motion” and mass “quantity of matter”. He considered matter indestructible and hence in an isolated system

dm/dt = 0.

The two force definitions are then identical and one wonders what insight prompted him to give two definitions. Today, motive force is the only exact definition. His three laws of motion may be stated in the terse relation:

In an isolated mechanical system, not influenced by
external forces:

7/ ∑dM/dt = 0

In Newton’s days the laws were seen as an expression of balancing forces; action and reaction. Today we interpret them as a law of conservation of momentum. This reading permits a smooth transition to the post-Einstein version of Newton’s mechanics, where dynamic force and kinetic energy have become unwieldy concepts.

In Newton’s time the concept of energy was poorly understood, if at all. The same latin word “vis” represented both energy and force. What we call kinetic energy, Leibnitz termed “vis viva”, i.e. living force.( By the same token, the term “heat” was not distinguished from “temperature”. It took later generations to sort it out.)

The term “energy”, -e-, denotes the ability of a system to perform work. Work and energy have the same dimensions. Both are scalars. Einstein’s axiom completes the list above:

8/ e = mc²

Solving these simultaneous differential equations in m and v yields the familiar:

9/ m = m₀(1 - v²/c²)^-1/2 = ƒm₀ (v² is the dot-product v.v)

The magnitude of the momentum vector is:

10/ |M| = m|v| = m₀(1/v² - 1/c²) ^-1/2

The kinetic energy is

11/ e = m₀c² (ƒ - 1)

Discussion

This relation, (9/), (identical to the Lorenz tranform) is deduced here directly from Einstein’s axiom. The axiom itself is presented as a first principle, without any carry-on baggage.

The relation tells us that, relative to the source of the force, “c” is an upper boundary for speed obtained by that force. It does not prevent us from treating v as an ordinary vector, regardless of its magnitude.

The relation, derived in this manner, applies strictly to bodies containing matter. and has no information about the dynamics and energy content of photons, except that for photons to escape, they must be ejected with velocity c, relative to the source.