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.