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Foundations of mechanics for 3D duration

The first edition of New Foundations for Classical Mechanics (1986) by David Hestenes included “Foundations of Mechanics” as Chapter 9. This was removed for lack of room in the second edition, but is available online as a pdf here. This length-time foundation may serve as a guide for the foundation of mechanics for duration-base. To do so requires terms and correspondences in addition to switching duration and length:

length (location) space → duration space, temporal order → stantial order, particle → transicle, location (position) → chronation, (object) body → subject body, instant → waypoint, clock → odometer, reference frame → reference timeframe.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for duration-base:

To begin with, we recognize two kinds of entities, transicles and subject bodies which are composed of transicles. Given a subject body, R, called a reference timeframe, each transicle has a geometrical property called its chronation with respect to R. We characterize this property indirectly by introducing the concept of a duration space. For each reference timeframe R, a duration space T is defined by the following postulates:

  1. T is a 3-dimensional Euclidean space.
  2. The chronation (with respect to R) of any transicle can be represented as a point in T.

The first postulate specifies the mathematical structure of a duration space while the second postulate supplies it with a physical interpretation. Thus, the postulates define a physical law, for the mathematical structure implies geometrical relations among the chronations of distinct transicles. Let us call it the Law of Stantial Order.

Notice that this law asserts that every transicle has a property called chronation and it specifies properties of this property. But it does not tell us how to measure chronation. Measurement is a separate matter, since it entails correspondence rules as well as laws. In actual practice the reference timeframe is often fictitious, though it is related indirectly to a physical subject. Our discussion is simplified by feigning that the reference timeframe is always a real subject body.

We turn now to the problem of formulating the scientific concept of a linear reference. We begin with the idea that a linear reference is a measure of motion, and motion is a change of time reference with respect to a given reference timeframe. The concept of a linear reference embraces two distinct relations: stantial order and temporally remote co-location. To keep this clear we introduce each relation with a separate postulate.

First we formulate the Law of Stantial Order:

The motion of any transicle with respect to a given reference timeframe can be represented as an orbit in a duration space.

This postulate has a semantic component as well as a mathematical one. It presumes that each transicle has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in a duration space. Recall that an orbit is a continuous, oriented curve. Thus, a transicle’s orbit in a duration space represents an ordered sequence of chronations. We call this order a stantial order, so we have attributed a distinct stantial order to the motion of each transicle.

To define a physical, linear reference scale as a measure of motion, we select a moving transicle which we call a transicle odometer We refer to each successive chronation of this transicle as a waypoint. We define the linear reference interval Δs between two waypoints by

Δs = cΔt,

where c is a positive numerical constant and Δt is the time reference interval of the odometer’s orbit between the two waypoints. Our measure of linear reference interval is thus related to the measure of duration in a duration space.

To use this linear reference scale as a measure for the motions of other transicles, we need to relate the motions of transicles at different chronations. The necessary relation can be introduced by postulating the

Law of Coincidence:

At every waypoint, each transicle has a unique chronation.

This postulate determines a correspondence between the points on the orbit of any transicle and points on the orbit of a transicle odometer. Therefore, every transicle orbit can be parametrized by a linear reference parameter defined on the orbit of a transicle odometer.

Note that this postulate does not tell us how to determine the chronation of a given transicle at any waypoint. That is a problem for the theory of measurement.

So far our laws permit orbits which are nondifferentiable at isolated points or even at every point. These possibilities will be eliminated by Newton’s laws which require differentiable orbits. We include in the class of allowable orbits, orbits which consist of a single time point across some linear reference interval. A transicle with such an orbit is said to be fixed with respect to the given reference timeframe through that linear reference interval. Of course, we require that the transicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a rigid subject body.

Note that the pace of a transicle is just a comparison of the transicle’s dischronment to the dischronment of a transicle odometer. The pace of a transicle odometer has the constant value 1/c = Δt/Δs, so the odometer moves uniformly by definition. In principle, we can use any moving transicle as an odometer, but the dynamical laws we introduce later suggest a preferred choice. Any moving transicle defines a periodic process, because it moves successively over time intervals of equal length. It should be evident that any real odometer can be accurately modeled as a transicle odometer. By regarding the transicle odometer as the fundamental kind of odometer, we make clear in the foundations of physics that the scientific concept of linear reference is based on an objective comparison of motions.

We now have definite formulations of a duration space, so we can define a reference system as a representation x for the possible chronation of any transicle at each linear reference r in some linear reference interval. Each reference system presumes the selection of a particular origin for a duration space and particular choices for the units of distance and duration, so each time reference and linear reference is assigned a definite numerical value. The term “reference system” is sometimes construed as a system of procedures for constructing a numerical representation of a duration space.

After we have formulated our dynamical laws, it will be clear that certain reference systems called facilial systems have a special status. Then it will be necessary to supplement our Law of Coincidence with a postulate that relates coincident points in different facilial systems. That is the critical postulate that distinguishes classical mechanics from special relativity, but we defer discussion of it until we are prepared to handle it completely. It is mentioned now, because our formulation of a duration space will not be complete until such a postulate is made.

It is convenient to summarize and generalize our postulates with a single law statement, the Zeroth (or Temporostancial) Law of Physics:

Every real movement has a continuous history in a duration space.

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