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Actual and possible motion

This post continues the topic posted here.

The action motion of a particle or rigid body may be measured by the scalar (or 1D) rate of motion, expressed as a speed or a pace. The numerator of a speed is the measured length or travel distance, and the denominator is the unit of time or the measured travel time. The pace is the inverse.

The possible motion of a particle or rigid body is limited to three dimensions of motion. This may be represented as a rate of motion in a Euclidean space of three-dimensions. The rate of motion is the speed or pace. With the direction of motion this is the velocity or allegrity.

If time (travel time) is held constant, this becomes a spatial 3D Euclidean geometry, commonly called “space”. If length (travel distance) is held constant, this becomes a temporal 3D Euclidean geometry, which may be called “time-world”.

The combination of space and time-world is a 6D Euclidean geometry of possible representations. This is not the extent of motion but the extent of the representation of motion, the next level of abstraction.

Motion vs. movement

The English words motion and movement are similar. They both have to do with “changing position or going from one place to another.” (Collins English Dictionary)

Then what’s the difference? Here are a few ways of putting it:

motion is used to describe physical properties, while movement is used to describe the qualities of motion. Ref.

motion doesn’t always imply a purpose, and movement usually does. Ref.

The difference is very fine. I would say that movement is déplacement d’un lieu à un autre [displacement from one place to another] whereas motion is le fait de ne pas rester immobile [not to stand still]. But usage and context are crucial. Ref.

People may not be consistent about it but for the purposes here they can be distinguished. Motion is the general term in kinetics, the study of motion. It says nothing about the purpose of a motion, or its origin and destination. Something just happens to change place.

However, movement includes some purpose, some origin and destination. A movement is a complete motion, from beginning to end. So movement would be preferred in the arts and social sciences and motion in the natural sciences.

Physics studies motion. Transportation studies movement. They may both speak about something changing position but there is a different perspective.

A movement is an entity, a thing, not just a change as a motion is. A motion can be studied abstractly but a movement is not fully abstract because it is an entity.

A body has its motion and a movement has its figure. A body is flesh-and-blood 3D, with motion only adding a thin 1D time perspective. A movement has 3D animation and life, with a figure only adding a thin 1D space perspective.

Bodies and motions

For kinematics and dynamics, one can begin with a body – something that takes space – and apply a motion to it. Or one can begin with a motion – something that takes time – and apply it to a body. In either case the result is a body and a motion – either a body with a motion or a motion with a body.

That, in a nutshell, is the difference between the spatiocosm and the tempocosm, that is, 3D space with 1D time and 3D time with 1D space. They are two ways of saying the same thing with a different emphasis.

Yes, a motion can be a “thing” – an entity of its own – the motion around a race track, for example. Applied to different contestants such a motion produces different timings. That is what the race is all about.

A body applies to different motions, too – a body such as a projectile, for example. When launched in different ways such a body travels different lengths, as in a contest for the longest length.

Compare the von Neumann computer architecture in which instructions and data share the same address space. Whether an address represents data or instructions depends only on the interpretation.

Different situations call for different perspectives. A race against time calls for a tempocosmic vew. A contest for distance calls for a spatiocosmic view.

Is there a combined perspective? Yes, in a relativistic sense, but it collapses to one or the other perspective as soon as a measurement is taken. If spatial position is measured, it’s a spatiocosmic view. If time and direction are measured, it’s a tempocosmic view.

Space-time duality

Space and time are dual concepts. They are complementary to one another as an inverse binary symmetry. They go together as space with time in classical physics or time with space or spacetime as in relativistic physics.

Parallel terms:

Space Time
3D space +1D time 3D time + 1D space
vector space + scalar time vector time + scalar space
spatiocosm tempocosm
reference frame reference timeframe
space position time position
particle moticle
a body with a motion a motion with a body
waypoint instant
odologe clock

For further details see here and other posts on this blog.

Linear, radial, and scalar space and time

In 3D space the distance between points is the magnitude of the displacement vector, which is a scalar length. In 3D time the duration between instants (time points) is the magnitude of the distimement vector, which is a scalar time.

Radial space is space with radial coordinates, 2D polar or 3D spherical, in which the distance from or to a point is the magnitude of the radial displacement. Radial time is time with radial coordinates, 2D polar or 3D spherical, in which the duration to an instant is the magnitude of the radial distimement. The radial space and time are “as the crow flies,” with the imaginary crow flying at a constant rate.

The distance between two points along a path (or curve) is the arc length (scalar space) between them. The duration between two instants along a path in time is the arc time (scalar time) between them. Each of these may be conceived of as one rectilinear dimension.

A measuring wheel can measure the arc length along the space path between points. An odologe can measure the arc time along the space path between events. An odologe is equivalent to a measuring wheel operating at a fixed rate. A stopwatch with a conversion speed is effectively an odologe.

The result of a measuring wheel operated throughout a region or network is a (spatial) linear referencing system (LRS or SLRS), which produces a set of physical wayposts or conceptual waypoints.

The result of an odologe operated throughout an event region or network is a temporal linear referencing system (TLRS), which produces a set of physical eventposts or conceptual eventpoints/eventicles. A visual or audible signal synchronized with an event would be an example of an eventpost. This might be a train conductor’s “all aboard!” or a signal at the stops of an automated transit system.

Terms for rates of motion

The scalar space of a motion is the arc length along the curve it traces out. The scalar time of a motion is the travel time along the route it traces out.

The time rate is “The rate at which something takes place over time.” The space rate is the rate at which something takes time over a route.

A quantity at an instant of time is instantaneous. A quantity at a point in space is punctaneous, from Latin punct(us) + (instant)aneous.

Speed is the time rate of motion, the scalar space per unit of scalar time. The speed at an instant of time is called the instantaneous speed, which equals the differential scalar space per differential scalar time or the magnitude of the instantaneous velocity.

Pace is the space rate of motion, the scalar time per unit of scalar space. The pace at a point of space is called the punctaneous pace, which equals the differential scalar space per differential scalar time or the magnitude of the punctaneous allegrity. A gradient is the space rate of change of a function or scalar field.

Displacement is the directed distance or vector difference between two points in three-dimensional space. Distimement is the directed duration or vector difference between two instants in three-dimensional time.

Velocity is the time rate of change of distimement, which consists of the speed and direction of motion. The average velocity is the displacement per scalar time of motion. The velocity at an instant of time is called the instantaneous velocity, and is the differential displacement per differential scalar time. The instantaneous velocity equals the time rate of change of the displacement.

Allegrity is the space rate of change of distimement, which consists of the pace and direction of motion. Allegrity is the rate of progress on a trajectory or path. The average allegrity is the distimement per scalar space of motion. The allegrity at a point of space is called the punctaneous allegrity, and is the differential distimement per differential scalar space. Allegrity is from allegr(o) + ity (cf. velocity).

Acceleration is the time rate of change of velocity. Modulation is the space rate of change of allegrity.

The first moment of mass is mass times distance. The momentum is the time rate of change of the first moment of mass. The first moment of vass is vass times duration. The space rate of change of the first moment of vass is the celentum, from Latin clim(a), slope + (mom)entum.

Power is the time rate of change of energy. Force is the space rate of change of energy.

Odologes

An odologe (o′∙do∙loje) a constant-rate length-measuring device symmacronized with a common waypoint. It is a new coinage from the Greek odo(s), way/path + (horo)loge, clock. In short, it is a clock that shows length or angle instead of time.

The simplest odologe takes time from a clock and multiplies it by a conversion speed to produce a length or angle. This is commonly done in relativity with c, the speed of light: ct equals time multiplied by the speed of light, resulting in a length. A device which output such a length would be an odologe.

Given a starting point the Moon could be considered an odologe that travels at the rate of 3,683 kilometers per hour, since that is its speed around the Earth. The Earth itself could be considered an odologe that travels at its orbital rate of 107,000 km/h. Another odologe is the distance of the Pioneer 10 spacecraft from the Earth and Sun, which is tracked here.

A circular analog clock is also an angular odologe of the angle made by each hand, as below:

Clock angles

Each minute is represented by 1/60 of a 360° circle, or 6° of arc. Each hour is represented by 1/12 of a circle, or 30° of arc.

There are 24*60 = 1440 minutes in a day, which equates to 360 = 1440/4 degrees, so the apparent mean speed of the Sun along the ecliptic is one degree every four minutes. The Sun provides an approximate angular odologe.

A kind of virtual odologe comes from using the typical speed for a vehicle in a metropolitan area to convert travel time into travel distance. So, for example, one might estimate that a traveler has progressed 20 kilometers since they have been gone a half hour and the typical speed is 40 km/h.

Another virtual odologe is an app that displays an odometer that increases at a constant rate, as illustrated below.

Why space and time are not different

Many differences are proposed between space and time. This post briefly indicates how all of them are a matter of convention, and so not real. For details, consult posts on this blog.

(1) There are three space dimensions but only one time dimension.

Directionality can be associated with either length (distance) or time (duration). 3D time is as legitimate as 3D space.

(2) The time is always changing (“flowing”) but space position need not change.

Time seems to “flow” because of its association with continual change, as of a river. But change can just as well be associated with length. Something floating down a river continually changes it length of travel. So a continual increase of length can be associated with change.

(3) There is an “arrow” of time, but not of space.

As in (2) above, if length is associated with change, then there is an “arrow” of length. Compare a vehicle’s odometer, which never decreases. An odologue shows length always increasing, like a clock.

(4) Time has a past, present, and future, but space does not.

Space includes where one was in the past, where one is now, and where one will be in the future.

(5) Travel takes place in space in all directions, but only in one direction in time.

Travel takes place in time in all directions since there is a travel time for each direction of travel. Space may be associated with the travel distance, which has one direction.

(6) Entropy grows with time, but not with space.

One can equally well define an entropy of space and show that it tends to increase with the distance of motion.

(7) Causality is ordered in time, not in space.

Causality is ordered in space as in time. What happens in one place is causally related to what happens in adjoining places, and on to other places.

(8) Causally connected events are timelike or lightlike, but cannot be spacelike.

In a 3D time and 1D space context, the properties of timelike and spacelike are interchanged.

Ten meanings of time

Carlo Rovelli’s “Analysis of the Distinct Meanings of the Notion of “Time” in Different Physical Theories” (Il Nuovo Cimento B, Jan 1995, Vol 110, No 1, pp 81–93) describes ten distinct versions of the concept of time, which he arranges hierarchically. Here are excerpts from his article:

We find ten distinct versions of the concept of time, all used in the natural sciences, characterized by different properties, or attributes, ascribed to time. We propose a general terminology to express these differences. p.81

… our aim is to emphasize the general fact that a single, pure and sacred notion of “Time” does not exist in physics. p.82

The real line is a traditional metaphor for the idea of time. Time is frequently represented by a variable t in R. The structure of R corresponds to an ensemble of attributes that we naturally associate to the notion of time. These are the following:
a) The existence of a topology on the set of the time instants, namely the existence of a notion of two time instants being close to each other, and the characteristic “one dimensionality” of time;
b) The existence of a metric. Namely the possibility of stating that two distinct time intervals are equal in magnitude. We denote this possibility as metricity of time.
c) The existence of an ordering relation between time instants. Namely, the possibility of distinguishing the past direction from the future direction;
d) The existence of a preferred time instant, namely the present, the “now”. p.83

In the natural language, when we use the concept of time we generally assume that time is one-dimensional, metrical, external, spatially global, temporally global, unique, directed, that it implies a present, and that it allows memory and expectations. The concept of time used in Newtonian physics is one-dimensional, metrical, external, spatially global, temporally global, unique, but it is not directed and it does not have a present. In thermodynamics, time has the additional property of being directed. Proper time along world line in general relativity is one-dimensional, metrical, temporally global but it is not external, not spatially global, not unique; on the other side, the time determined by a matter clock is one-dimensional, metrical, but not temporally global, an so on. p.87

… the notion of present, of the “now” is completely absent from the description of the world in physical terms. This notion of time can be described by the structure of an affine line A. p.88

… our list does not include the possibility of considering a non-metric but directional notion of time. p.89

Table I. [without the fourth column]

Time concept Attributes Example
time of natural language memory brain
time with a present present biology
thermodynamical time directional thermodynamics
Newtonian time uniqueness Newton mechanics
special relativistic time being external special relativity
cosmological time space global proper time in cosmology
proper time time global world line proper time
clock time metricity clocks in general relativity
parameter time 1-dimensional coordinate time
no time none quantum gravity

… our hypothesis concerning time is that the concepts of time with more attributes are higher-level concepts that have no meaning at lower levels. p.91

If this hypothesis is correct, then we should deduce from it that most features of time are genuinely meaningless for general systems. p.91

… we suggest that the very notion of time, with any minimal characterization, is likely to disappear in a consistent theory that includes relativistic quantum-gravitational systems. p.91

… the concept of time, with all its attributes, is not a fundamental concept in nature, but rather that time is a progressively more specialized concept that makes sense only for progressively more special systems. p.92

Space and time involution

J. C. C. McKinsey, A. C. Sugar and P. Suppes (hereafter MSS) wrote “Axiomatic foundations of classical particle mechanics”, (Journal of Rational Mechanics and Analysis, v.2 (1953) p.253-272), which is also described in Suppes’ Introduction to Logic (Van Nostrand, New York, 1957), pp.291-322 (see here). It is only a partial axiomatization of Newtonian mechanics but is sufficient to present an involution of mechanics below.

An involution in mathematics is a function that is its own inverse, which means if it is repeated the output is the input, that is, f(f(x)) = x. The involution here is the interchange of spatial and temporal quantities along with the inversion of mass. We start with MSS:

MSS system has six primitive notions: P, T, m, s, f, and g. P and T are sets, m is a real-valued unary function defined on P, s and g are vector-valued functions defined on the Cartesian product P × T, and f is a vector-valued function defined on the Cartesian product P × P × T. Intuitively, P corresponds to the set of particles and T is to be physically interpreted as a set of real numbers measuring elapsed times (in terms of some unit of time, and measured from some origin of time). m(p) is to be interpreted as the numerical value of the mass of p ∈ P. sp(t), where tT, is a 3-dimensional vector which is to be physically interpreted as the position of particle p at instant t. f (p, q, t), where p, qP, corresponds to the internal force that particle q exerts over p, at instant t. And finally, the function g(p, t) is to be understood as the external force acting on particle p at instant t. (Anna & Maia p,9)

We define MSS´ by the following involution: interchange P and Q; p and q; T and S; m and n=1/m; s and t; f and k; g and ; particle and moticle; instant and point; mass and vass. Then the explanation is:

MSS´ system has six primitive notions: Q, S, n, t, k, and . Q and S are sets, m is a real-valued unary function defined on Q, t and  are vector-valued functions defined on the Cartesian product Q × S, and k is a vector-valued function defined on the Cartesian product Q × Q × S. Intuitively, Q corresponds to the set of moticles and S is to be physically interpreted as a set of real numbers measuring scalar space (in terms of some unit of length, and measured from some origin point). n(q) is to be interpreted as the numerical value of the vass of qQ. tq(s), where sS, is a 3-dimensional vector which is to be physically interpreted as the position of moticle q at point s. k(q, p, s), where q, pQ, corresponds to the internal surge that moticle p exerts over q, at point s. And finally, the function (q, s) is to be understood as the external surge acting on moticle q at point s.

The corresponding axioms are as follows:

A1 Q is a non-empty, finite set.
A2 S is an interval of real numbers.
A3 If qQ and sS, then tq(s) is a 3-dimensional vector (tq(s) ∈ℜ³) such that d²tq(s)/ds² exists.
A4 If qQ, then n(q) is a positive real number.
A5 If p, qQ and sS, then k(p, q, s) = −k(q; p; s).
A6 If p, qQ and sS, then tq(s) × k(p, q, s) = –tp(s) × k(q, p, s).
A7 If p, qQ and sS, then n(q) d²tq(s)/dt² = ΣpQ k(p, q, s) + ℓ(q, s).

These axioms generate a dual to Newtonian mechanics. A5 corresponds to a weak dual version of Newton’s Third Law: to every surge there is always a countersurge. A6 and A5, correspond to the strong dual version of Newton’s Third Law. A6 establishes that the direction of surge and countersurge is the direction of the line defined by the coordinates of moticles p and q. A7 corresponds to the dual of Newton’s Second Law.

MSS show that mass is independent of the remaining primitive notions of their system. Because of this, its dual could be defined differently. It was thought best to take the inverse of mass, called vass, for the involution.