Kinematics, the geometry of motion, studies the positions of geometric objects parameterized by time. This is a 3D space with functions representing the path or trajectory as the locus of places occupied by points. It has a dual mathematics of 3D time with functions representing the course of motion as the locus of times occupied by events. Below is an introduction to both, following the exposition in Principles of Engineering Mechanics: Kinematics by Millard Beatty Jr.

1.3 Motion and Particle Path

To locate an object in space, we need a reference system. The only reference we have is other objects. Therefore, the physical nature of what we shall call a reference frame is an assigned set of objects whose mutual distances do not change with [dis]time – at least not very much. …

We define a three-dimensional Euclidean reference frame φ as a set consisting of a point O of space, called the origin of the reference frame, and a vector basis {e_i} ≡ {e₁, e₂, e₃}. That is, φ = {O; e_i}. We shall require for convenience that the basis is an orthonormal basis, i.e., a triple of mutually perpendicular unit vectors.

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This continues the previous post here.

The parametric equation for a circular helix around the x₁-axis with radius r and slope b/a (or pitch 2πb) is x₁(t) = bt, x₂(t) = r cos(t), x₃(t) = r sin(t). Its arc length equals t · sqrt(r² + b²).

The parametric equation for a circular helix around the t₁-axis with radius q and slope c/q (or pitch 2πc) is t₁(x) = cx, t₂(x) = a cos(x), t₃(x) = q sin(x). Its arc length equals x · sqrt(q² + c²).

The linear motion along the x₁-axis measures length, s, with velocity b. The circular motion around the x₁-axis measures time, t, as an angle. The linear motion along the t₁-axis measures length, w, with velocity c. The circular motion around the t₁-axis measures length, x, as an angle.

(1) If the circular motion around the x₁-axis is independent, it measures time, t, as an angle. If the linear motion along the x₁-axis is dependent, it measures length, s, as a parameter, and the axis is a space axis, x_s.

(2) If the linear motion along the x₁-axis is independent, it measures length, x, as a parameter, with velocity b. If the circular motion around the x₁-axis is dependent, it measures time, t, as an angle, and the axis is a time axis, x_t.

(3) If the motion is helical, that is both circular and linear, then if it is measured by length, it forms a curve in 3D space. If it is measured by time, it forms a curve in 3D time.

Galilean transformation with independent time: x_t´ = x_t, x_s´ = x_s – v_x. x_t, y_s´ = y_s, , z_s´ = z_s.

Simplified notation: t´ = t, x´ = x – vt, y´ = y, z´ = z.

Galilean transformation with independent space: x_s´ = x_s, x_t´ = x_t – u_x. x_s, y_t´ = y_t, , z_t´ = z_t.

Simplified notation: s´ = s, t´ = t – ux, t₂´ = t₂, t₃´ = t₃.

This post continues from the previous post here.

All motion is axial. It is a principle of kinematics that every motion is composed of the simple motions of translation and rotation. These simple motions are either along an axis (translation) or around an axis (rotation).

Here is a table of these two simple motions:

Length measurement is the ratio of the target length to a standard unit of length, such as one metre. Time measurement is the ratio of the target angle to a standard unit of angle, such as one revolution.

Space is a manifold of three dimensions of translation. Time is a manifold of three dimensions of rotation, which are three axes. These comprise six axes of motion, or six degrees of freedom.

The space and time exchange postulate means that the kinematic variables of translation and rotation are symmetric in the equations of motion. That is, a dual equation of motion is formed by exchanging space and time variables.

As described in the previous post here, the three dimensions of motion are axes for traveling along (length) or revolving around (time).

A measure of motion may be either (1) dependent on the the target motion, or (2) independent of the target motion. A measure that is independent is either available prior to or separately from the target motion. For example, an independent measure may be determined by agreement, such as the length of a race, or it may measure another motion, such as the motion of a clock, which is then correlated with the target motion.

A standard clock measures time because it measures rotations around an axis as an angle. A length clock measures rotations about an axis as a length. With constant rates of rotation constant, there is a fixed ratio between the two kinds of clock.

A device that measures its own internal motion may be called an autometer. A clock is an example of an autometer. The internal motion of an autometer can be correlated with a target motion. For clocks this is called synchronization. For a length clock this is called symmacronization.

An odometer is a measurement device that depends on its target motion. The standard odometer measures length of travel. A time odometer, or trip-timer, measures time of travel. A trip-timer is a stopwatch that is on only while the target motion takes place. If there is a stop in the target motion, then the trip-timer also stops. So the trip-timer measures time of motion rather than elapsed time.

A device that measures a quantity of motion need not be attached to the moving body. The theory of relativity deals with the remote measurement of quantities of motion. A device that is attached to the moving body produces proper measures such as proper length or proper time.

As a thought experiment, consider a rifle bullet, conceived of as an inertial projectile, fired at a target. Let the bullet itself be a source of measurement units: there is the length of the bullet and the rotation of the bullet. The extent of the motion of the bullet to the target could then in principle be measured as a number of bullet lengths and a number of rotations.

Length is the number of bullet lengths. Time is the number of bullet rotations. Thus length is essentially a linear measure and time is essentially a rotational measure. Length is generalized as the correspondence of the motion to a linear object, a rigid rod or ruler, which forms the basis of space. Time is generalized as the correspondence of the motion to a rotational object, a clock, which forms the basis of (abstract) time.

A rifle bullet provides a way to conceive of motion coordinates. Consider individually labeled rifle bullets continually fired from a common origin toward three orthogonal directions. The coordinates of a particular motion are then the labels on the coordinates that correspond to the motion. This means there are three pairs of coordinates: two for each bullet. These axes are the six degrees of freedom.

Motion conceived as a length function of time means that each for each rotational coordinate there corresponds its paired length coordinate. Motion conceived as a time function of length means that for each length coordinate there corresponds one paired rotation coordinate.

In conclusion, there are three dimensions of mobility. There are three dimensions for each measure of the extent of motion, which totals six dimensions. For ordinary purposes, the three dimensions of motion are sufficient, with space and time kept separate. But for science, which seeks a unified treatment, space and time should be united into six dimensions.

Victor Yakovenko has a derivation (see here) of the Lorentz Transformation (LT) in which he uses “only the equivalence of all inertial reference frames and the symmetries of space and time.” Because of the use of (spatial) reference frames and velocity, this is not completely symmetric. As we have seen, there is a dual Lorentz Transformation. Let us follow Yakovenko’s derivation but with reference timeframes and legerity (matrix forms omitted).

1) Let us consider two inertial reference timeframes P and P´. The reference timeframe P´ moves relative to P with legerity u along the t₁ ≡ t axis. We know that the coordinates t₂ and t₃ perpendicular to the legerity are the same in both reference timeframes: t₂ = t₂´ and t₃ = t₃´. So, it is sufficient to consider only transformation of the coordinates x and t from the reference timeframe P to x´ = f_x(x; t) and t´ = f_t(x; t) in the reference timeframe P´.

From translational symmetry of space and time, we conclude that the functions f_x(x, t) and f_t(x, t) must be linear functions. Indeed, the relative distances between two events in one reference timeframe must depend only on the relative distances in another timeframe:

 t´₁ – t´₂ = f_t(x₁ – x₂, t₁ – t₂),     x´₁ – x´₂ = f_x(x₁ – x₂, t₁ – t₂).          (1)

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Although there are many experimental methods available to measure the speed of light, the underlying principle behind all methods [is] the simple kinematic relationship between constant velocity, distance and time given below:

c = D / t                     (1)

In all forms of the experiment, the objective is to measure the time required for the light to travel a given distance. (Ref.)

From the perspective of the experimenter, light is an object whose speed is to be determined. Even though the distance traversed is fixed, it is placed in the numerator because this speed is to be compared with the speeds of other objects. For the same reason the quantity to be measured, time, is placed in the denominator.

But if we take the perspective of the experiment, of what is measured, then the fixed distance is the independent variable, which is placed in the denominator. The dependent variable is the time, which is placed in the numerator, so the pace of light is measured:

c´ = t / D                     (2)

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We follow the presentation of Guy Moore, formerly of McGill University, online here.

The Doppler effect is the phenomenon we have all noticed, that a sound produced by a moving source, or which you hear while you are moving, has its perceived frequency shifted.

Spacetime (3+1)

If a source of sound makes a sudden bang (pressure pulse) at time 0, then the location in space of the pressure pulse in relation to time will look like successive concentric circles. For a source moving to the right, letting out a series of “bangs,” the location of the successive pressure peaks will be closer together on the right and further apart on the left.

Now remember that a sound is just a series of pressure peaks, which are tightly separated in time, which is the period of the wave. Therefore, instead of thinking of the sound waves as due to “bangs,” you can think of them as pressure peaks in a periodic sound wave. In front of the moving object the peaks are closer together. That means that the wave length is shorter, which means it is a higher frequency wave. Behind, the peaks are farther apart, meaning that it is a longer wavelength sound, at a lower frequency. This is the gist of the Doppler effect.

Let us now actually calculate the size of the effect. Suppose a sound source is moving right at you, at velocity v. At time 0, it emits a pressure peak. At time Δt, it emits a second pressure peak. If its distance from you at time 0 was x, its distance from you at time Δt was x – vΔt (it is nearer, since it is moving towards you).

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