iSoul In the beginning is reality

Knowledge and repetition

Consider the distinction between repeatable events from unrepeatable events. Repeatable events includes events that have repeated or may be repeated at will (as in a laboratory) or may possibly repeat in the future. Unrepeatable events are events that are very unlikely to repeat or are impossible to repeat. It is said that science only studies repeatable events, and it can be argued that history is the study (science) of unrepeatable events – not that it excludes repeatable events but that it focuses on unrepeatable events.

“Nature” could be defined as the realm of repeatable events. Then natural science would be the study of nature or repeatable events. Those events that are unrepeatable would be left to historians but ignored by natural scientists. But could such scientists rightly study the past while ignoring unrepeatable events? Ignorance of unrepeatable events would be a limitation and a defect. We would not expect historians to ignore repeatable events, so why expect scientists to ignore unrepeatable events?

We may well expect events that only involve inanimate nature are repeatable in some way. But are all events with living beings repeatable? The position of naturalism says, Yes. But at some point we need to say, No, at least some living beings have free will (or whatever you want to call it) so that their actions may be unrepeatable, and thus beyond the purview of a science of repeatable events.

Knowledge of repeatable and unrepeatable events may need different methodologies to address both kinds of events but it could not ignore either kind without bias. We need both the study of history, with its unrepeatable events, and the study of science, and its repeatable events, as independent disciplines. The synthesis of science and history would require a different discipline, perhaps called “scihistory” or “histence”, that would balance the input of each discipline with the other.

Posts on space and time chronologically, updated

I previously listed posts on space and time chronologically here. This is a chronological list that includes the posts since then, starting with the most recent (with hyperlinks):

Outline of spacetime symmetry paper
Work and energy, exertion and verve
Circular orbits
Foundations of mechanics for time-space
Distance, duration, and angles
Center of vass
Equations of motion in space-time and time-space
Derivation of Newton’s second law
Clock race
Centripetal prestination
Motion equations revised
Gravitation and levitation theories
Simple harmonic motion
1D space + 3D time again
Measuring mass
Numbers large and small
Dynamic time-space
Four rates of motion
No motion as zero speed or pace
Simple motion in space and time
Observability of the rotation of the earth
Places in time
Conventions of here and now
Places and events
Four space and time dimensions
2D space + 2D time
Event-structure metaphors
Space and time standards
Space and time from the beginning
Dual differential physics
Time and linear motion
Time and circular motion
Gravity with dependent time
Sun clocks
Inverse terminology
Passenger kinematics
Physics for travelers
Non-uniform motion
Uniform motion
Two ways to symmetry
More equations of motion
Relating space and time
Parallel equations of motion
Corresponding equations of motion
Glossary of time-space terms
“Synchronizing” space
Characteristic limits
Minimum speeds
Modes and measures
Direction in three-dimensional time, part 3
Problems in mechanics, part 2
Measurement by motion
6D as two times 4D
Transformations for one or two directions
Travel time and temporal displacement
Galilei doesn’t lead to Lorentz
Transformations for time and space
Six dimensions of space-time
Time scale maps
A new geometry for space and time
Why time is three dimensional
Necessary and possible dimensions
Geometric and temporal unit systems
Time conventions
Direction in three-dimensional time, part 2
Dimensions of space and time
Terminology for time-space
Newtonian laws of motion in time-space
Phases of a 3D time theory
Paceometer
Problems in mechanics, part 1
Equations of motion in time-space
Conservation of prolentum
Dynamics for 3D time
Flow of independent variables
Switching space and time
Lorentz transformation for 3D time
Space and time expanded
Pace of light
Terminology for space and time, part 3
Synchrony conventions
Consciousness of space and time
Lorentz transformations and dimensions
Fixed sizes and rates in space and time
Lorentz and co-Lorentz transformations
Galilean and co-Galilean transformations
Relativity of time at any speed
Motion science basics
Flow of motion
3D time + 1D space, pace, and lenticity
Three dimensional clock
3D time in ancient culture
Relativity at any speed
1D space and 3D time
Characteristic speeds
6D space-time collapses into 4D
Direction in three-dimensional time, part 1
Terminology for space and time, part 2
Lorentz transformation in any direction
Superluminal Lorentz transformation again
The physics of a trip
Measuring movement
Total time
Dimensions of movement
Time on space and space on time
Dual Galilei and Lorentz transformations
Measurement of space and time
Invariant interval check
Six dimensional space-time
Two one-way standard speeds
Movement and dimensions
Insights on the complete Lorentz transformation
Subluminal and superluminal Lorentz transformations
Complete spatial and temporal Lorentz transformations
Limits of the Lorentz transformation
Lorentz for space & time both relative?
Absolute vs relative space, time, and dimension
Complete Lorentz group
Complete Lorentz transformation
Four perspectives on space and time
Change flows
Three arguments for 3D time
Variations on a clock
Conversion of space and time
Time and memory
Time in the Bible
Temporal and spatial references
Perspectives on space and time
Homogeneity and isotropy
Multidimensional time in physics
Multidimensional time in transportation
Angles in space and time
Basis for the symmetry of space and time
Lorentz without absolutes
Optimizing travel time routes
Different directions for different vectors
Claims about time, updated
Modes of travel
Lorentz for space and time
Galilei for space and time
The speed of spacetime
Representations of space and time
Travel in space and time
Proof of three time dimensions
Velocity with three-dimensional time
Dimensions of dimension
Space, time, and spacetime
Time and distance clocks
Actual and default speeds
Time at Mach 1
Centers of time measurement
Directional units
Cycles and orbits
Converting space and time
Actual and potential time and space
Defining space and time
Equality of space and time
Kinds of relativity
Symmetric laws of physics
Diachronic and synchronic physics
Measurement of space and time
Lorentz with 3D time
Time defined anew
Lorentz interpreted
Lorentz generalized
Transportation and physics
Average spacetime conversion
Galileo revised
Movement and measurement
Distance without time
Measurement
Velocity puzzle
Bibliography of 3D time and space-time symmetry
Symmetries and relativities
Distance, duration and dimension
Coordinate lattices
Independent and dependent time
Claims about time
Reality and relativity
What is single-value time?
Parametric time and space
Speed and its inverse
Symmetry of space and time
An introduction to co-physics, part 2
An introduction to co-physics, part 1
Terminology for space and time, part 1
Direction and units of magnitude
Six dimensional spacetime
Duals for Galilean and Lorentz transformations
Geometric vectors in physics
Speeds and velocities
Direction and dimension
No change in time per distance
The flow of time and space
Is time three-dimensional?
Is space one-dimensional?
Time in spacetime
Space, time and causality
Mechanics in multidimensional time
Measures of speed and velocity
Homogeneity and isotropy of time
Multidimensionality of time
Space, time, and arrows
Arrow of tense
Duality of space and time

Outline of spacetime symmetry paper

This is an outline of an article on “The Symmetry of Space and Time”. I’ll update it as needed and add links to the parts as they are written.

0.0 Abstract

1.0 Introduction

1.1 Examples of multi-dimensional time, ancient and modern
1.2 Reference to related work
1.3 Overview of the paper

2.0 Simple motion in 1+1 dimensions (space and time)

2.1 Distance and duration
2.2 Symmetry of space and time

3.0 Motion in 3+1 (space-time) is symmetric with motion in 1+3 dimensions (time-space)

3.1 Classical Kinematics
3.1.1 Angles and turns in 3D
3.1.2 Speed and pace, velocity and celerity
3.1.3 Equations of motion

3.2 Classical Dynamics
3.2.1 Mass and vass, momentum and celentum
3.2.2 Equations of motion
3.2.3 Newtonian gravitation in time-space

4.0 Motion in 3+3 dimensions (spacetime)

4.1 Mechanics in spacetime (3+3), reduction of 3D into 1D
4.2 Lorentz transformations in 3+3, invariant interval for 3+3

5.0 Conclusion

6.0 References

~

Claims:

  1. Physics (mechanics) begins with the study of local simple motion in 1+1 dimensions
  2. Physics (mechanics) may be done in either space-time (3+1) or time-space (1+3)
  3. Physics (mechanics) is within a spacetime (3+3) framework
  4. Time may be seen as having 3-dimensions just as well as space
  5. Time is duration with direction. That is, time is a vector variable similar to a space vector (a distance with a direction). Duration is measured by a standard rate of change
  6. The magnitude of time is that which is measured by a stopwatch, similar to length
  7. Replacing time with its negation produces a duration in the opposite direction. It does not reverse time or switch past and future
  8. Rates require a scalar in the denominator, which can be either space (distance) or time (duration)
  9. The spatial and temporal perspectives are complementary opposites. Time and space are symmetric with one another, and so may be conceptually interchanged
  10. Both time and space have continuous symmetries of homogeneity and isotropy
  11. Minkowski spacetime may be expanded to six dimensions, three for time and three for space. That is, the invariant distance is: (ds)² = (c dtx)² + (c dty)² + (c dtz)² – (drx)² – (dry)² – (drz
  12. Overall claim: space and time are symmetric

Work and energy, exertion and verve

Here we show the work and energy in the linear motion of a particle in space-time (see J.M. Knudsen and P.G. Hjorth’s Elements of Newtonian Mechanics, 1995, p.51). Consider a particle of mass m moving along the r axis so all quantities are scalars. Newton’s second law is then

mr/dt² = F, or

m dv/dt = F,

with mass m, force F, and velocity v. Multiply both sides by v = dr/dt:

m (dv/dt) v = F (dr/dt), or

mv dv = F dr := dW,

where W is called the work done by the force F over the segment dr. Define T as

T = mv²/2,

which is called the kinetic energy of the particle. Then

dT = dW.

That is, the change in the kinetic energy of the particle over the segment dr equals the work done by the force F.

If F = F(r) does not depend on time, then define the potential energy U = U(r) through

dU(r) := –dW = –F(r)dr.

That is, the change in the potential energy U(r) over the segment dr is equal to minus the work done by the external force F. Since

dT = –dU(r),

and upon integrating,

T + U(r) = E,

where the constant E is called the total mechanical energy of the system.


Here we show the exertion and verve in the linear motion of an eventicle (point vass) in time-space. Consider an eventicle of vass ℓ moving along the t axis so all quantities are scalars. Newton’s second law for time-space is then

ℓ d²t/dr² = Γ, or

ℓ du/dr = Γ,

with vass ℓ, surge Γ, and celerity u. Multiply both sides by u = dt/dr:

ℓ (du/dr) uΓ (dt/dr), or

ℓu duΓ dt := dX,

where X is called the exertion done by the surge Γ over the time segment dt. Define V as

V = ℓu²/2,

which is called the kinetic verve of the eventicle. Then

dV = dX.

That is, the change in the kinetic verve of the eventicle over the segment dt equals the exertion done by the surge Γ.

If Γ = Γ(t) does not depend on position, then define the potential verve Y = Y(t) through

dY(t) := –dX = –Γ(t)dt.

That is, the change in the potential verve V(t) over the time segment dt is equal to minus the exertion done by the external surge Γ. Since

dV = –dY(t),

and upon integrating,

V + Y(t) = Z,

where the constant Z is called the total mechanical verve of the system.

 

Science in the center

There are many different musical temperaments that have been used to tune musical instruments over the centuries. They all have their advantages and disadvantages. But there is one musical temperament that is optimally acceptable: the equal temperament method in which the frequency interval between every pair of adjacent notes has the same ratio. This produces a temperament that is a compromise between what is possible and what is agreeable to hear.

Science faces many situations such as the challenge of musical temperament. Conventions and methods need to be adopted and there are multiple options, each with their advantages and disadvantages. There are those who promote one method and those who promote another method, often the opposite method. Should science pick one and force everyone to conform? Or should science find a compromise of some sort?

There is a way in the middle that is a compromise between extremes and alternatives. It is a conscious attempt to avoid extremes and biases, and seek a solution that is the most acceptable to all. This is science in the center, a science that minimizes bias. Although it might be called “objective,” that obscures the fact that it is a conscious choice.

I previously wrote about the need for a convention on the one-way speed of light. Science of the center would avoid the bias toward one direction of light and choose a one-way speed that is in the middle between all the possible speed conventions. This is the Einstein convention, which is part of his synchronization method.

Science in the center includes not biasing classifications either toward “lumping” or “splitting.” Nor should explanations of behavior be biased toward “nature” or “nurture.” The particulars of each case should determine the outcome, not a preference for one side or the other. If there’s any default answer, it’s in the center between such extremes.

Occam’s razor is understood to prescribe qualitative parsimony but allow quantitative excess. This is as biased as its opposite would be: to prescribe quantitative parsimony but allow qualitative excess. Science in the center would avoid the bias that each of these has by prescribing a compromise: there should be a balance between the qualitative and the quantitative. Neither should be made more parsimonious than the other. All explanatory resources should be treated alike; none should be more abundant or parsimonious than any other. I’ve called this the New Occam’s Razor, and it is an example of science in the center.

Circular orbits

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions:

  1. Each planet orbit the Sun in a spatially circular path.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is constant.

Let’s follow the exposition given in Elements of Newtonian Mechanics by J.M. Knudsen and P.G. Hjorth (Springer, 1995), starting on page 6. Because the speed is constant, the acceleration follows the equation for uniform circular motion derived previously:

a = / Rs,

in which v is the speed and a is the acceleration. We have from the definition of speed:

v = S / T = 2πRs / T.

Elimination of v from these equations leads to

aRs / , or

a ∝ S / .

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

.

Combining this with the equation for acceleration yields

a ∝ 1 / Rs², or

a ∝ 1 / , or

a ∝ 1 / T4/3, or

a ∝ 1 / Rt4/3.

Inserting the first acceleration into Newton’s second law leads to:

Fm / Rs²,

with force, F, and mass, m. The force is directed toward the Sun, with a magnitude inversely proportional to the square of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite force from each planet toward the Sun. Which is to say:

FM / Rs².

for mass, M. The combined law of gravitation is thus:

F = GmM/Rs²,

for some constant G.

Now consider gravity at the surface of the Earth. Newton showed that the gravitational force of a body would be the same if the mass were concentrated at its center, so the above equation can be used with the radius of the Earth, ρ²:

F = GmM/ρ² = mg,

with g as the acceleration of gravity on Earth. Then

g = GM/ρ².

If the known values of G, M, and ρ are inserted into this equation, the result is g = 9.8 m/s².


Let us now modify this derivation so that distance is the independent variable instead of time. This is better called a law of levitation since it is naturally directed toward the smaller mass. We have then three propositions:

  1. The Sun orbits each planet in a temporally circular path.
  2. The Sun is at the center of vass of its orbit.
  3. The pace of the Sun is constant.

Because the pace is constant, the prestination follows the equation for uniform circular motion derived previously:

b = / Rt,

in which u is the pace and b is the prestination. Again, distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

We have from the definition of pace:

u = T / S = 2πRt / S.

Elimination of u from these equations leads to

bRt / , or

bT / , or

bT / Rs².

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

, or

Rt².

Combining this with the equation for prestination yields

b ∝ 1 / Rt1/3, or

b ∝ 1 / T1/3, or

b ∝ 1 / S1/2, or

b ∝ 1 / Rs1/2.

Inserting the latter prestination into Newton’s second law in the form of surge, Γ, gives:

Γ ∝ ℓ / Rs1/2,

with surge, Γ, and vass, ℓ. The surge is directed away from the Sun, with a magnitude inversely proportional to the square root of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite surge toward each planet from the Sun. Which is to say:

Γ ∝ L / Rs1/2,

for vass, L. The combined law of levitation is thus:

Γ = HℓL/Rs1/2,

for some constant H. Since this is a function of the inverse square root instead of the inverse square, it is more sensitive to changes in Rs, as with a body in radial motion.

Consider levity at the surface of the Earth, using the above equation with the radius of the Earth, ρ:

Γ = HℓL/ρ1/2 = nh,

with h as the prestination of levity on Earth. Then

q = HL/ρ1/2.

The values for H, L, and ρ may be inserted into this equation to determine the value of h.

Foundations of mechanics for time-space

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 space in the second edition, but is available online as a pdf here. This space-time foundation may serve as a guide for the foundation of mechanics for time-space. To do so requires terms and correspondences in addition to switching space and time:

space → time space, time → space length, particle → eventicle, body → time body, instant → spot, clock → odologe, 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 time-space:


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

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

The first postulate specifies the mathematical structure of a time 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 time positions of distinct eventicles. Let us call it the Law of Temporal Order.

Notice that this law asserts that every eventicle has a property called time position and it specifies properties of this property. But it does not tell us how to measure time position. 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 time body. Our discussion is simplified by feigning that the reference timeframe is always a real time body.

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

First we formulate the Law of Spatial Order:

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

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

To define a physical, space length scale as a measure of motion, we select a moving eventicle which we call an eventicle odologe. We refer to each successive time position of this eventicle as a spot. We define the space length interval Δs between two spots by

Δs = cΔt,

where c is a positive numerical constant and Δt is the arc time of the odologe’s orbit between the two spots. Our measure of space length is thus related to the measure of duration in time space.

To use this space length scale as a measure for the motions of other eventicles, we need to relate the motions of eventicles at different time positions. The necessary relation can be introduced by postulating the

Law of Coincidence:

At every spot, each eventicle has a unique time position.

This postulate determines a correspondence between the points on the orbit of any eventicle and points on the orbit of an eventicle odologe. Therefore, every eventicle orbit can be parametrized by a space length parameter defined on the orbit of an eventicle odologe.

Note that this postulate does not tell us how to determine the time position of a given eventicle at any spot. 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 through some space length interval. An eventicle with such an orbit is said to be fixed with respect to the given reference timeframe through that space length interval. Of course, we require that the eventicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a rigid time body.

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

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

After we have formulated our dynamical laws, it will be clear that certain reference systems called alacrital 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 alacrital 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 time space and space length 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 Temporospatial) Law of Physics:

Every real time body has a continuous history in time space and space length.

Distance, duration, and angles

Let’s follow the orbit of a particle or the route of a vehicle as a curvilinear function with associated directions at every point. Measurement produces travel distance r, travel time t, with directions θ and φ. The directions may be considered as functions of either travel distance or travel time: θr, φr, θt, or φt. There are accordingly four possibilities:

(r, t, θr, φr), (r, t, θt, φt), or (t, r, θr, φt), or (t, r, θt, φr).

The latter two may be made equal by a change of convention for measuring the angle. These may be represented rectilinearly as:

(t, rx, ry, rz), (r, tx, ty, tz), (rw, rx, ty, tz), or (tw, tx, ry, rz).

The latter two may be made equal by a change of convention for the axes.

Three possibilities remain: (3D space + 1D time), (1D space + 3D time), or (2D space + 2D time).

An example of the third possibility would be a traveler who measured their horizontal angle relative to magnetic north and their vertical angle relative to the sun. Since magnetic north is (approximately) fixed, it serves to measure the horizontal angle spatially. Since the sun’s position continually changes, it serves to measure the vertical angle temporally. The result is (2+2) with (r, θr) and (t, φt).

Or one could do the opposite and measure the horizontal angle temporally, as with a sundial, and the vertical angle spatially, as with a theodolite. The result is (2+2) with (t, θt) and (r, φr).

If both angles are measured relative to a fixed point, then the result is (3+1) or (t, r, θr, φr). If both angles are measured relative to a moving point, then the result is (r, t, θt, φt). The moving point should be moving at a constant rate, or at least a constant acceleration.

If three coordinates are measured relative to a fixed axis, then the result is (1+3) or (t, rx, ry, rz). If three coordinates are measured relative to a rotating axis, then the result is (r, tx, ty, tz). The moving axis should be moving at a constant rate, or at least a constant acceleration.

The potential reality of (r, t, θr, φr, θt, φt) collapses to one of the possibilities above in the act of measurement. The potential reality of (rx, ry, rz, tx, ty, tz) collapses to one of the rectilinear possibilities above in the act of measurement.

Center of vass

The vass is to time (duration) as the mass is to space (distance). As noted before here, the vass can be measured by a similar procedure as the mass. The mass and vass are inverses with opposite uses.

center of mass

The center of mass is the point that two or more particles (point masses) are balanced (or one large mass is balanced). For two particle masses, m1 and m2 that are located at points x1 and x2, respectively:

Center of Mass (CM) = (m1x1 + m2x2)/(m1 + m2),

which is the weighted arithmetic mean with the masses as the weights. This is similar to the momentum, in which the velocity is weighted by the mass: mv.

The center of vass is the point in time that two or more particle vasses are balanced. For two particle vasses, ℓ1 and ℓ2 that are located at points in time t1 and t2, respectively:

Center of Vass (CV) = (ℓ1t1 + ℓ2t2)/(ℓ1 + ℓ2) = ((t1/m1 + t2/m2)/(1/m1 + 1/m2)),

which is the weighted arithmetic mean with the vasses as the weights. Compare the celentum, in which the celerity is weighted by the vass: ℓu.

In order to generalize this, let’s use the derivation of the center of mass, as in Knudsen and Hjorth’s Elements of Newtonian Mechanics, chapter 9. Start with the time position vector, T, to find the center of vass for a system of particle vasses:

L Tcv = Σ ℓi ti,

where ℓi and ti are the vass and the time position vector of the ith particle vass, and L = Σ ℓi is the total vass of the system. Then differentiate with respect to space length to get

L Tcv´ = Σ ℓi ti´= Σ qi := Q,

where the total linear celentum of the system is denoted Q. In other words, the total linear celentum Q of a system of particle vasses is the same as that of a particle vass with vass L moving the the celerity of the center of vass. This is also stated as

Q = L ucv,

where ucv is the celerity of the center of vass.

Equations of motion in space-time and time-space

First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = t, position = r, initial position = r(t0) = r0, velocity = v, initial velocity = v(t0) = v0, v = |v| = speed, and acceleration = a.

First equation of motion

v = ∫ a dt = v0 + at

Second equation of motion

r = ∫ (v0 + at) dt = r0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (rr0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (rr0)).


Next, here is a derivation of the time-space equations of motion, in which prestination is constant. Let position (displacement) = r, time = t, initial time = t(r0) = t0, celerity = u, initial celerity = u(r0) = u0, u = |u| = pace, and prestination = b.

First equation of motion

u = ∫ b dr = u0 + bt

Second equation of motion

t = ∫ (u0 + br) dr = t0 + u0r + ½br²

Third equation of motion

From u² = uu = (u0 + br) ∙ (u0 + br) = u0² + 2r(bu0) + b²r², and

(2b) ∙ (tt0) = (2b) ∙ (u0r + ½br²) = 2r(bu0) + b²r² = u² ‒ u0², it follows that

u² = u0² + 2(b ∙ (tt0)).