Chapter 43
Law of Conservation of Momentum
Motion and Inertial Coherence
Author: Mark Anthony Higgins
Revised Date: 23 February 2026
Chapter 43
Overview
The conservation of momentum is the law of reciprocal memory — the assurance that when one body changes its motion, another changes in equal and opposite measure.
In classical form, it states that within a closed system with no external forces, total momentum remains constant:
$$ \sum_i \mathbf{p}_i = \text{constant}, \qquad \frac{d}{dt}\sum_i \mathbf{p}_i = 0. $$
In coherence terms, it represents the continuity of curvature exchange between interacting boundaries.
When coherence is perfect (), the field transmits momentum flawlessly — no loss, no delay, no asymmetry.
When coherence is partial (), the exchange becomes imbalanced — energy leaks into other modes (heat, deformation, radiation), and momentum symmetry fades.
🔹 1. Classical Momentum and System Closure
For each body:
$$ \mathbf{p}_i = m_i \mathbf{v}_i, \qquad \mathbf{F}_i = \frac{d\mathbf{p}_i}{dt}. $$
In a closed system, all internal forces are paired:
$$ \sum_i \mathbf{F}_i = 0. $$
Hence,
$$ \frac{d}{dt}\sum_i \mathbf{p}_i = 0. $$
and total momentum is conserved.
This expresses Newton’s Third Law at the system level —
each action and reaction cancel as equal and opposite curvature flows, leaving the total geometry unchanged.
🔸 2. Coherence and Momentum Flow
In coherence geometry, momentum is a boundary flow of curvature — the directional memory of energy within a system.
When two bodies interact, momentum is exchanged through their shared boundary; energy and curvature reflect symmetrically.
This reflection is governed by the boundary coherence index :
$$ \mathbf{p}_2' = -\,I_B\,\mathbf{p}_1'. $$
Thus, momentum conservation describes how energy and curvature redistribute within a boundary memory, not merely a transfer of “mass × velocity.”
🔹 3. Elastic and Inelastic Coherence
| Interaction Type | Coherence (IB) | Behaviour | Example |
|---|---|---|---|
|
Elastic
|
𝐼 𝐵 = 1
|
Perfect momentum
and energy reflection |
Ideal collision,
photon recoil |
|
Inelastic
|
0 < 𝐼 𝐵 < 1 |
Partial energy retention, |
Real material collision
|
|
Dissipative
|
𝐼 𝐵 → 0 |
Total momentum
asymmetry, energy lost to field |
Absorptive impact
or radiation |
Elastic interactions preserve total field coherence — every impulse mirrored in equal curvature.
Inelastic interactions show how partial coherence breaks symmetry yet still respect global conservation through redistribution.
🔸 4. External Impulse and Open Systems
If an external force acts on the system, momentum changes by impulse:
$$ \Delta \mathbf{p}_{\mathrm{total}} = \int \mathbf{F}_{\mathrm{ext}}\,dt. $$
This opens the boundary — coherence is no longer closed, and total system momentum shifts.
However, the underlying field remains conservative: the external region absorbs the counter-momentum.
Momentum is never destroyed — it simply leaves one coherence loop to join another.
🔹 5. Feedback Law for Coherence Decay
Loss of coherence corresponds to gradual relaxation of boundary alignment:
$$ \dot{I}_B = -\lambda(1 - I_B). $$
As coherence fades, the reciprocal symmetry between interacting bodies weakens — momentum exchange becomes less instantaneous.
This is the physical root of drag, viscosity, and radiative recoil:
momentum is still conserved, but redistributed across temporal feedback delay.
🧠 Summary
The Law of Conservation of Momentum is the geometry of perfect feedback.
It is not a coincidence of forces, but a continuity of curvature:
energy flowing across boundaries while preserving the total coherence of spac
Momentum conservation, in the HST sense, is the reciprocal memory of motion —
the universe remembering every interaction it performs.
No motion is lost; it is only rephrased — another boundary, another echo of coherence.
