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Chapter 46
Torque and Rotational Coherence

Motion and Inertial Coherence
Author: Mark Anthony Higgins
© Mark Anthony Higgins 11 October 2025
Revised Date: 23 February 2026
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Flow Algebra 
flow "Torque and Rotational Coherence" v1.0

# ---------- PURPOSE ----------
# Expresses rotational dynamics through the principle of coherence.
# Torque is treated as the curvature feedback linking angular acceleration and rotational energy.
# Perfect coherence (IB = 1) → torque transfers angular momentum without loss (rigid rotation).
# Reduced coherence (IB < 1) → partial energy transfer (friction, deformation, slippage).
# The flow unifies τ = Iα and L = Iω under the same coherence framework used for linear motion.

# ---------- PARAMETERS ----------
params:
  units: "SI"
  IB: 0.0..1.0 = 1.0         # rotational coherence index (1 = perfect rigidity)
  I: positive                 # moment of inertia [kg·m²]
  τ_ext: real                 # external torque applied [N·m]
  α: real                     # angular acceleration [rad/s²]
  ω_i: real                   # initial angular velocity [rad/s]
  ω_f: real                   # final angular velocity [rad/s]
  θ: real                     # angular displacement [rad]
  loss_coeff: 0.0             # coherence decay rate [s^-1]
  dt: 0.0?                    # timestep [s]
  include_feedback: true

# ---------- STATE ----------
state Rotation:
  L: real                     # angular momentum [kg·m²/s]
  τ_net: real                 # net torque [N·m]
  ΔE_rot: real                # change in rotational energy [J]
  E_loss: real                # rotational energy loss [J]
  IB_rate: real               # rate of coherence change
  balance_error: real         # |τθ - ΔE_rot| difference [J]
  rotation_state: str         # "stable", "dissipative", or "accelerating"

# ---------- INVARIANTS ----------
invariants:

  # Torque and angular acceleration relation
  Rotation.τ_net = params.IB * params.I * params.α

  # Angular momentum L = Iω
  Rotation.L = params.I * params.ω_f

  # Work done by torque (rotational analogue of W = F·s)
  W_rot = Rotation.τ_net * params.θ

  # Change in rotational kinetic energy
  Rotation.ΔE_rot = 0.5 * params.I * (params.ω_f^2 - params.ω_i^2)

  # Perfect coherence → τθ = ΔE_rot
  if params.IB == 1:
    Rotation.balance_error = abs(W_rot - Rotation.ΔE_rot)
    Rotation.rotation_state = "stable"
  else:
    Rotation.E_loss = (1 - params.IB) * W_rot
    Rotation.ΔE_rot = params.IB * W_rot
    Rotation.rotation_state = "dissipative"

  # Coherence decay
  if params.loss_coeff > 0 and params.include_feedback:
    params.IB = params.IB - params.loss_coeff * (1 - params.IB) * (params.dt or 1)
    params.IB = clamp(params.IB, 0, 1)
    Rotation.IB_rate = -params.loss_coeff * (1 - params.IB)

# ---------- TRANSFORMS ----------
transforms:

  # Update angular velocity (α = τ/I)
  propagate_rotation (when params.dt > 0):
    params.α = Rotation.τ_net / params.I
    params.ω_f = params.ω_i + params.α * params.dt
    Rotation.L = params.I * params.ω_f
    Rotation.ΔE_rot = 0.5 * params.I * (params.ω_f^2 - params.ω_i^2)

  # Coherence feedback correction
  coherence_feedback:
    correction = -0.001 * Rotation.balance_error
    params.IB = clamp(params.IB + correction, 0, 1)

# ---------- MONITORS ----------
monitors:
  energy_balance:
    if params.IB == 1:
      assert abs(Rotation.τ_net * params.θ - Rotation.ΔE_rot) < 1e-9
      comment: "Perfect rotational coherence — all torque converts to energy."
    else:
      comment: "Partial coherence — torque partially dissipated as heat or deformation."
  coherence_bounds:
    assert 0 <= params.IB <= 1
  angular_momentum_check:
    assert Rotation.L >= 0

# ---------- SCENES ----------
scenes:

  perfect_rotation:
    given params:
      I: 2.0
      α: 5.0
      θ: 10.0
      IB: 1.0
      ω_i: 0.0
      ω_f: 10.0
    expect:
      Rotation.τ_net = 10.0
      Rotation.ΔE_rot = 100.0
      comment: "Perfect coherence — rotational work equals energy change."

  partial_rotation:
    given params:
      I: 2.0
      α: 5.0
      θ: 10.0
      IB: 0.9
      ω_i: 0.0
      ω_f: 10.0
    expect:
      Rotation.ΔE_rot = 0.9 * τθ
      Rotation.E_loss = 0.1 * τθ
      comment: "Partial coherence — 10% energy lost to internal friction."

  frictional_wheel:
    given params:
      I: 3.0
      α: 4.0
      θ: 5.0
      IB: 0.85
      loss_coeff: 0.05
      dt: 1.0
    enable:
      - propagate_rotation
    expect:
      params.IB ↓
      Rotation.E_loss > 0
      comment: "Torque decays due to rotational friction — coherence fades over time."

  gyroscopic_precession:
    given params:
      I: 5.0
      ω_i: 100.0
      α: 0.0
      IB: 0.95
      θ: 0.1
    expect:
      Rotation.τ_net small
      Rotation.rotation_state = "stable"
      comment: "Gyroscopic stability — near-perfect coherence, minimal energy loss."

# ---------- NOTES ----------
notes:
  - **Torque–Energy Relation:** \( \tau \theta = \Delta E_{\text{rot}} \)
  - **Rotational kinetic energy:** \( E_{\text{rot}} = \tfrac{1}{2} I \omega^2 \)
  - **Torque law:** \( \tau = I \alpha \)
  - **Angular momentum:** \( L = I \omega \)
  - **Power in rotation:** \( P = \tau \omega \)
  - **Coherence index (IB):**
    - \(I_B = 1\): perfect rigidity — full torque transfer.
    - \(I_B < 1\): internal slip — rotational energy lost to heat or vibration.
  - **Feedback law:** \(\dot{I_B} = -\lambda(1 - I_B)\)
  - **Interpretation:** Torque represents curvature tension in angular motion — the feedback between force and rotational displacement.
  - **Dynamic meaning:** Rotation is sustained curvature coherence; torque acts as the mechanism that restores or disrupts that coherence.
  - **Analogy:** Just as linear force bends velocity in a line, torque bends angular momentum around a center of curvature.
  - **Visualization:** Show torque vector perpendicular to rotation plane, energy flowing between τθ and ΔE_rot, with faded arrows indicating loss when IB < 1.
Overview

Torque is the curvature analogue of force — the rotational tension that bends inertia around a center instead of through a line.
It measures how momentum flows across a circular path, maintaining angular coherence between the applied force and the resulting motion.

In classical terms:

$$ \tau = I\alpha, \qquad L = I\omega. $$

where τ\tau is torque, II is moment of inertia, α\alpha is angular acceleration, and LL is angular momentum.

In coherence terms, torque is the feedback of curvature memory — the rotational counterpart of linear force.
When coherence is perfect (IB=1I_B = 1), torque and motion are phase-locked: every newton-metre of applied curvature tension converts fully into angular acceleration and energy.
When coherence fades (IB<1I_B < 1), part of the torque diffuses — converted into friction, heat, or internal deformation rather than rotation.

🔹 1. Classical Form — Torque as Angular Force

Work in rotation mirrors linear work:

$$ W_{\text{rot}} = \tau \theta. $$

and the corresponding energy change:

$$ W_{\text{rot}} = \Delta E_{\text{rot}} = \frac{1}{2} I (\omega_f^2 - \omega_i^2). $$

In perfect rotational coherence, these two quantities balance exactly — no loss, no phase drift:

$$ \tau \theta = \Delta E_{\text{rot}}. $$

This expresses a pure curvature transfer: torque applied over an angular distance creates rotational energy with 100% fidelity.

🔸 2. Coherence Index and Energy Partition

In real systems, coherence is rarely perfect.
Some of the applied torque disperses into local vibrations or thermal motion — micro-curvature noise within the structure.
The coherence factor IBI_B quantifies this retention:

$$ \Delta E_{\text{rot}} = I_B\,(\tau \theta), \qquad E_{\text{loss}} = (1 - I_B)\,(\tau \theta). $$

  • IB=1I_B = 1: rigid coherence — all torque converts to rotation.
  • IB<1I_B < 1: internal slip or damping — partial torque loss.
  • IB=0I_B = 0: no coherent rotation — all input energy dissipated.

The coherence field thus mediates how efficiently angular curvature is transmitted across a material or structure.

🔹 3. Angular Momentum as Rotational Memory

Angular momentum encapsulates rotational coherence:

$$ L = I\omega. $$

It acts as rotational inertia’s memory — a conserved curvature current.
When no external torque acts:

$$ \frac{dL}{dt} = 0. $$

This is the rotational analogue of Newton’s First Law:
the system continues spinning with constant angular momentum until an external curvature (torque) interacts with it.

When torque is applied, it alters the geometry of this memory:

$$ \tau = \frac{dL}{dt}. $$

The faster LL changes, the greater the curvature stress required to maintain coherence.

🔸 4. Coherence Loss — Friction and Structural Lag

Loss in rotational coherence corresponds to micro-asymmetries in torque distribution —
small phase differences between applied force and structural response.
This can be captured dynamically as:

$$ \dot{I}_B = -\lambda(1 - I_B). $$

Here, λ\lambda represents how quickly a system forgets its rotational memory due to internal drag or damping.

  • In a rigid flywheel, λ≈0\lambda \approx 0; IB≈1I_B \approx 1 — motion remains stable and coherent.
  • In a viscous or elastic medium, λ>0\lambda > 0; IB<1I_B < 1 — torque spreads unevenly and energy decays as heat.
  • In fluid-like systems, λ→1\lambda \to 1; IB→0I_B \to 0 — rotational coherence is nearly lost.

This gradual loss represents the conversion of organized rotation into disordered motion — curvature transitioning from macro to micro scale.

🔹 5. Power and Rotational Feedback

Rotational power defines the rate at which torque performs work:

$$ P = \tau \omega. $$

In full coherence, PP maps directly to the rate of energy transfer;
in partial coherence, only a fraction (IBI_B) contributes to usable rotation:

$$ P_{\text{eff}} = I_B\,\tau\,\omega. $$

The remainder ((1−IB)τω(1 - I_B)\tau \omega) manifests as internal heating or vibrational damping.
This shows torque as a dynamic coherence agent — continuously shaping and realigning the geometry of motion.

🧠 Summary

Torque is not merely a twisting force — it is rotational curvature in motion.
It transmits energy between angular and linear domains through coherence feedback.
When the system’s geometry is fully aligned (IB=1I_B = 1), rotation conserves its energy perfectly.
When coherence decays (IB<1I_B < 1), angular memory diffuses — motion transforms into internal noise, but the total energy remains conserved across scales.

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