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Motion Equations

Equations for circular/harmonic and spiral/helical motion in a length and duration manifold is given in a pdf here. The kinematic equations below have different forms depending on whether the motion is linear or angular (rotational) and whether length and duration are vectors or scalars. They are also given in a pdf here. Circular/harmonic motion equations are here.

Parallel Equations of Motion
RA Gillmann, 2020-02-06 Displacement + Time Dischronment + Stance Spatial Angles + Time Temporal Angles + Stance
Stance/Distance | Time/Distime Displacement: s

Time | duration: t

Dischronment: t

Stance | length: s

Spatial angle: θ = s/r

Temporal arc: t

Temporal angle: φ = t/q

Spatial arc: s

Radius | Period Spatial radius

r = S/(2π) = qv

Temporal radius

q = T/(2π) = ru

Angular velocity

ω = φ/t = κ/u = 2πf = 2π/T = dθ/dt = dt/dφ = v/r = 1/q

Wavenumber (angular lenticity)

κ = θ/s = ω/v = 2πh = 2π/λ = dθ/ds = ds/dφ = u/q = 1/r

Velocity | Lenticity | Wavenumber Cross/Tangential velocity

v = ds/dt = r×ω = r/q = S/T

Cross/Tangential lenticity

u = dt/ds = q×κ = q/r = T/S

Acceleration | Retardation Radial acceleration
a = v2/r =2 = v/q = r/q²
Radial retardation
b = u2/q =2 = u/r = q/r²
Angular acceleration
α = dω/dt = aT/r
Angular retardation
β = dκ/ds = bT/q
Tangential acceleration
a = ω×v = Tdv/dt = rα
Tangential retardation
b = Tdu/ds = qβ
Wavelength | Period λ = S = 2πr = 2πvq T = 2πq = 2πur λ = S = 2π/κ = 1/h T = 2π/ω = 1/f
Revolutions | Repetitions
Frequency | Circuncy
Revolutions

N = θ/(2π)

Repetitions

Z = φ/(2π)

Temporal frequency
f = ω/(2π) = 1/T
Spatial frequency (circuncy)
h = κ/(2π) = 1/λ
Displacement | Dischronment s = s0 + vt t = t0 + ux θ = θ0 + ωt φ = φ0 + κs
First Equation of Motion v = v0 + at u = u0 + bs ω = ω0 + αt κ = κ0 + βs
Second Equation of Motion s = s0 + v0t + ½at² t = t0 + u0s + ½bs² θ = θ0 + ω0t + ½αt2 φ = φ0 + κ0t + ½βs2
Third Equation of Motion v² = v0² + 2a∙(s s0) u² = u0² + 2b∙(t t0) ω² = ω0² + 2α∙(θ θ0) κ² = κ0² + 2β∙(φ φ0)
Inertia | Facilia Mass (linear inertia): m = 1/n Etherance (linear facilia): n = 1/m Angular inertia: I = mr2 Angular facilia: J = nq2
Momentum | Fulmentum Momentum: p = mv Fulmentum: q = nu Angular momentum: L = Iω Angular fulmentum: Γ = Jκ
Kinetic Energy and Lethargy Kinetic energy: EK = ½mv2 Kinetic lethargy: LK = ½nu2 Angular energy EA = ½Iω2 Angular lethargy LA = ½Jκ2
Newton’s Second Law Force: F = ma = dp/dt Release: R = nb = dq/ds Torque: τ = Iα = s × F Strophence: σ = Jβ = t × R
Work | Repose Linear work: W = Fs Linear repose: Y = Rt Angular work: W = τθ Angular repose: Y = σφ
Power | Placidity Linear power: P = Fv Linear placidity: Z = Ru Angular power: P = τω Angular placidity: Z = σκ