We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.

A transicle is defined as a moving object of insignificant size. The motion of a transicle of vass n at the chronation t is governed by Newton’s Second Law for time-space, R = nb or, more precisely,

R(t; t′) = h′           (1.1)

where R is the release which, in general, can depend on both the chronation t as well as the lenticity t′, and h = nt′ is the fulmentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to stance x. Equation (1.1) reduces to R = nb if n′ = 0. But if n = n(x), then the form with h′ is correct.

General theorems governing differential equations guarantee that if we are given t and t′ at an initial stance x = x₀, we can integrate equation (1.1) to determine t(x) for all x (as long as R remains finite). This is the goal of classical dynamics.

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This is an update and expansion of the post here.

Here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time = t, location = x, initial location = x(t₀) = x₀, velocity = v, initial velocity = v(t₀) = v₀, speed = v = |v|, and acceleration = a.

First equation of motion

v = ∫ a dt = v₀ + at

Second equation of motion

x = ∫ (v₀ + at) dt = x₀ + v₀t + ½at²

Third equation of motion

From v² = v ∙ v = (v₀ + at) ∙ (v₀ + at) = v₀² + 2t(a ∙ v₀) + a²t², and

(2a) ∙ (x ‒ x₀) = (2a) ∙ (v₀t + ½at²) = 2t(a ∙ v₀) + a²t² = v² ‒ v₀², it follows that

v² = v₀² + 2(a ∙ (x ‒ x₀)), or

v² − v₀² = 2a ∙ x, with x₀ = 0.

Here is a derivation of the time-space equations of motion, in which retardation is constant. Let stance = x, time (chronation) = t, initial time = t(x₀) = t₀, lenticity = w, initial lenticity = w(x₀) = w₀, pace w = |w|, and retardation = b.

First equation of motion

w = ∫ b dx = w₀ + bx

Second equation of motion

t = ∫ (w₀ + bx) dx = t₀ + w₀x + ½bx²

Third equation of motion

From w² = w ∙ w = (w₀ + bx) ∙ (w₀ + bx) = w₀² + 2x(b ∙ w₀) + b²x², and

(2b) ∙ (t ‒ t₀) = (2b) ∙ (w₀x + ½bx²) = 2x(b ∙ w₀) + b²x² = w² ‒ w₀², it follows that

w² = w₀² + 2(b ∙ (t ‒ t₀)), or

w² − w₀² = 2b ∙ t, with t₀ = 0.

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What is a clock? it is a device that measures time, but what are the essentials of a clock? I submit these are the essentials of a clock:

(1) A clock requires a uniform motion. Because only the kinematics (not the dynamics) are significant, a uniform rotation is acceptable. But because the result will be represented as a line – a timeline or time axis – a linear uniform motion has a more direct connection with what is measured, so let us take the first essential as a uniform linear motion.

(2) In order for clocks to be measuring alike, it is necessary that there be a standard rate for all clocks. In addition, clocks should have a standardized beginning point, so that clocks are interchangeable.

(3) A clock requires a pointer which indicates the present time on a time scale as it moves at the standard uniform rate. This would be the hands and dial on a common analogue clock. On a linear clock it is a part whose position in motion is interpreted as the present value of time. The pointer and scale are essentials of a clock.

Furthermore, a clock must be interpreted as showing the present time of the observer’s rest frame.

Figure 1

All the essentials of a clock can be represented by a frame in standard uniform motion relative to the observer’s rest frame. In that case, a clock should be definable in terms of frames of reference: one rest frame and one frame in uniform motion relative to the observer’s rest frame, as in the following.

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