There are three axes (dimensions) of motion with six degrees of freedom. There are two metrics of motion: a length metric and a duration metric. The length metric is the magnitude of the vector between two points, and is called distance. The duration metric is the magnitude of the vector between two instants, and is called distime.

If one conceives of this as two 3D metric geometries of motion, then there is a 3D space geometry with a distance metric and a 3D time geometry with a distime metric. If the speed of light is an absolute conversion between distance and distime (which is essentially Einstein’s second postulate of special relativity), then there is one 6D spacetime metric geometry.

A 3D space coordinate system is built from an origin point and three orthogonal axes with a distance metric. A 3D time coordinate system is built from an origin instant and three orthogonal axes with a distime metric. A 6D spacetime coordinate system is built from an origin event, three space coordinates, and three time coordinates converted to lengths. An equivalent 6D spacetime coordinate system has three time coordinates and three space coordinates converted to durations.

A stance line represents two opposite linear motions with a constant rate (i.e., inertial motions). The positive direction represents distances to events diverging away from the origin point. The negative direction represents distances to events converging toward the origin point (i.e., destination). Apart from motion a point has a distance but its sign is ambiguous. A stance line represents the stance or scalar space of an odologe.

A time line represents two opposite straight motions with constant rate (i.e., inertial motions). The positive direction represents distimes to events diverging from the origin instant. The negative direction represents distimes to events converging toward the origin instant (i.e., destination). Apart from motion an instant has a distime but its sign is ambiguous. A time line represents the time or scalar time of a clock.

Events may be ordered by the stance or the time. Events ordered by stance are macronological. Events ordered by time are chronological. All events that are equal distances (equidistant) from the origin point are simulstanceous with it. All events that are equal distimes (equidistimed) from the origin instant are simultaneous with it.

Here we expand 4D Galilean spacetime into 6D Galilean spacetime, based on section 1.3 Galilean spacetime of The Geometry of Relativistic Spacetime: from Euclid’s Geometry to Minkowski’s Spacetime by Jacques Bros (Séminaire Poincaré 1 (2005) 1 – 45).

[p.3] We start with a representation space whose points are interpreted as the “physical events”. Any motion of a particle which is physically possible between two given events A and B is represented by a certain world-line with end-points A and B. There is an absolute orientation of such worldline, which can be called its “time-arrow”: its physical meaning is that one of the end-point events, e.g. B, is in the future of the other one A.

[p.6] From the viewpoint of mathematical physics, the use of geometry in more than three dimensions turns out to be necessary, if one wishes to represent phenomena whose description necessitates more than three independent quantities. A typical example is the six dimensional space R_ab⁶ = R_a³ × R_b³ of the positions (a; b) of pairs of material points (or pointlike particles) in mutual interaction. Trajectories of such pairs are represented by curves in R⁶, described in terms of a parameter t by equations of the form a = a(t); b = b(t).

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This post is about the conservation of (space) energy and time energy. I wrote about the conservation of fulmentum here. See also the post on Work, effort and energy.

First, here is a derivation of the conservation of (space) energy from classical physics:

The law of the conservation of (space) energy states that the total (space) energy in an isolated system remains constant over time (distime). The total (space) energy over an arbitrary length of distime, Δt, is constant. Let the total (space) energy at two times be E₁ and E₂. Then:

(E₂ – E₁)/Δt = 0.

Since the total energy equals the kinetic space energy (KSE) plus the potential space energy (PSE), we have

(KSE₂ + PSE₂ – KSE₁ – PSE₁)/Δt = 0

= (KSE₂ – KSE₁)/Δt + (PSE₂ – PSE₁)/Δt = 0

= (ΔKSE – ΔPSE)/Δt.

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The outline below is also available in pdf form here.

Spiral/Helical Motion

A helix is the geodesic of a cylinder; if we develop the cylinder on which the helix is traced, the helix becomes a straight line. Radius r (or a or R or A); velocity v, arc length s, arc time, w, pitch length P; pitch time, M; pitch angle α; pitch time angle β

Constants

v = |v| = √(r² + b²)          s = t √(r² + b²)

u = |u| = √(q² + c²)         w = x √(q² + c²)

Pitch and slope

pitch length, P = 2πb     slope, P/S = b/r

pitch time, M = 2πc       time slope, M/T = c/q

Pitch angle

α = atan(P/S) = atan(b/r)       β = atan(M/T) = atan(c/q)

Arc length of one winding    L = √(P² + S²)

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The outline below is also in pdf form here.

Angular speed (velocity) and angular pace (legerity)

• Speed, v = Δs/Δt, Pace, u = Δt/Δs so u = 1/v and v = 1/u except if u or v are zero
  • Zero speed: no motion but time changes because time is independent
  • Zero pace: no motion but length changes because length is independent

Circular motion in space and time

distance, x; distime, t; radius r (or a or R or A); period radius q; circumference S = 2πr = wavelength λ; period T = 2πq = wavetime μ; angular velocity, v; angular legerity, u; arc length, s; arc time, w

• Circle in space
  • space angle θ, arc length s, radius r
  • angle in space: θ ≡ s/r; r = s/θ; s = rθ; 1/θ = r/s; 1/r = θ/s
  • angular time rate: ω ≡ θ/t; t = θ/ω; θ = ωt; 1/ω = t/θ; 1/t = ω/θ
• Circle in time
  • time angle φ, arc time w, period radius q
  • angle in time, φ ≡ w/q; q = w/φ; w = qφ; 1/φ = q/w; 1/q = φ/w
  • angular space rate: ψ ≡ φ/x; x = φ/ψ; φ = ψx; 1/ψ = x/φ; 1/x = ψ/φ

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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.