Soumendra Nath Thakur
ORCiD: 0000-0003-1871-7803
December 31, 2025
Extended Classical Mechanics' Pre-relativistic mass-energy equivalence Principle.
1. Classical Force Laws:
Newton’s Second Law
F = m a
Force is the cause of acceleration of inertial mass.
2. Newton’s Law of Gravitation:
Fɢ = G (M m/r²)
This gives the force acting on a mass (m) due to a source mass (M).
3. Dynamical Equivalence:
Set gravitational force equal to inertial force:
m a = G (M m/r²)
Cancel (m):
a = GM/r²
This shows that gravitational acceleration is independent of the test mass.
4. Gravitational Field:
g(r) = GM/r²
So the gravitational force becomes
Fɢ = mg(r)
5. Physical Causal Chain:
M → g(r) → a → F
This is exactly what Newtonian gravity means physically:
Mass produces a field, the field produces acceleration, and acceleration produces force.
6. Classical Total Mechanical Energy:
Eₜₒₜₐₗ = PE + KE
This states that the energy of a classical system is the sum of its stored (potential) and motion (kinetic) energy.
For motion in a gravitational field,
Eₜₒₜₐₗ = mgh + ½mv²
This provides the exact classical baseline, against which the ECM generalization is being built.
7. Equivalence of Inertial and Gravitational Mass — ECM Formulation:
In classical mechanics, inertial mass and gravitational mass are empirically identical:
mɪₙₑᵣₜᵢₐₗ = mɢᵣₐᵥᵢₜₐₜᵢₒₙₐₗ
mɪ = mɢ
This Weak Equivalence Principle states that the mass that resists acceleration is the same mass that produces and experiences gravity.
Classical physics treats this equality as fundamental, without explaining why it holds.
8. The ECM Conceptual Breakthrough:
Classical mechanics assumes mass is a single primitive quantity:
mɪ = mɢ = m
This works because classical theory does not resolve how energy, inertia, and gravity are stored or generated within mass.
Extended Classical Mechanics (ECM) rejects this assumption.
In ECM, mass has internal structure.
9. Mass Structure in ECM:
In ECM, gravitational and inertial behaviour arises from the effective mass:
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mɢ
where
• Mᴍ is matter mass (existence field)
• −Mᵃᵖᵖ is Negative Apparent Mass (NAM) generated by manifested potential energy
−Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ
Thus, gravitational mass is not just matter mass — it is matter plus manifested energy field.
10. Classical Mechanics as a Special Case of ECM:
When no manifestation occurs,
−ΔPEᴇᴄᴍ = 0 → −Mᵃᵖᵖ = 0
so ECM collapses to
Mᵉᶠᶠ = Mᴍ
which gives
mɪ = mɢ
Therefore, the equivalence principle is not fundamental — it is the zero-manifestation limit of ECM.
11. What Classical Physics Cannot See:
Classical mechanics unknowingly treats
m = Mᴍ + (−Mᵃᵖᵖ)
as a single undifferentiated constant.
ECM reveals that:
• gravity
• inertia
• kinetic energy
• dark mass effects
all arise from the hidden NAM field, generated by −ΔPEᴇᴄᴍ.
Classical physics contains:
• no −Mᵃᵖᵖ
• no ΔMᴍ
• no mass–energy exchange
• no internal mass structure
So it reduces to:
Mɢ → m, Mᴍ → m, −Mᵃᵖᵖ → 0
which automatically yields
mɪ = mɢ
12. Physical Meaning:
Classical mechanics treats the entire ECM mass–energy structure as if it were already collapsed into a single constant (m).
That is why Newtonian gravity works — but it does not know why.
ECM provides the missing physics:
Classical mass = matter mass + hidden NAM field
but classical theory cannot separate them.
13. Final ECM Statement:
Mɢ = Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ), −Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ
In classical mechanics:
−ΔPEᴇᴄᴍ → 0 → Mɢ = Mᴍ
hence
mɪ = mɢ
ECM shows that this equality hides a deeper mass-energy structure that becomes visible only when motion, gravity, and manifestation occur.
14. Physical interpretation:
Classical mechanics treats the entire mass–energy structure of ECM as if it were already “pre-collapsed” into a single constant (m).
That is why classical gravity works — but it does not know why.
ECM tells the missing physics:
Classical mass matter mass + hidden NAM field
but classical theory cannot separate them.
15. Conclusion:
In classical mechanics, mɪ = mɢ because classical physics assumes −Mᵃᵖᵖ.
ECM reveals that this equality hides a deeper structure:
m = Mᴍ + (−Mᵃᵖᵖ),
which becomes visible only when motion and gravity exist.
16. ECM Extension of the Classical Force Law:
Extended Classical Mechanics (ECM) is built on the principle that kinetic energy is not independent of mass. Instead, kinetic energy arises from the redistribution of mass through the entropic release of stored potential energy,
−ΔPEᴇᴄᴍ.
In ECM, motion is powered when a portion of the system’s stored potential energy PEᴇᴄᴍ is converted into a manifested mass component,
−ΔPEᴇᴄᴍ ≡ −Mᵃᵖᵖ
which generates effective mass
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)
and it is this effective mass that participates in inertia, gravity, and acceleration.
Thus, what appears in classical physics as “kinetic energy” is, in ECM, the dynamic manifestation of mass created by potential-energy release.
17. Classical Force Law (Newtonian Interface)
F = M a
This equation remains exactly valid in ECM — but ECM now reveals what the symbol (M) really contains.
In classical mechanics, (M) is treated as a single primitive quantity.
In ECM, that same (M) is the collapsed form of
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ)
where
• Mᴍ is matter mass (existence field),
• −Mᵃᵖᵖ is Negative Apparent Mass (NAM) generated by −ΔPEᴇᴄᴍ.
Therefore, the classical force law becomes, in ECM meaning:
F = Mᵉᶠᶠ a = (Mᴍ + (−Mᵃᵖᵖ)a,
showing that force acts on the combined matter-mass plus manifested energy-mass.
18. The ECM Bridge Behind Newton’s Law:
The Newtonian equation
F = M a
corresponds in ECM to the causal chain
F ⟶ −ΔPEᴇᴄᴍ ⟶ −Mᵃᵖᵖ ⟶ Mᵉᶠᶠ ⟶ a.
Thus, force does not directly produce acceleration. It first releases potential energy, which generates NAM, which modifies effective mass, and only then produces motion.
This is the hidden physics behind Newton’s law.
19. Newtonian Limits (Recovered Exactly)
For classical fixed mass (M):
a ∝ F
For fixed force (F):
a ∝ 1/M
These relations are not altered in ECM.
They are simply the collapsed limit where
−Mᵃᵖᵖ ⟶ 0, Mᵉᶠᶠ ⟶ Mᴍ,
so that ECM reduces to
F = Mᴍ a,
which is ordinary Newtonian mechanics.
What ECM adds?
Newton gives the result.
ECM provides the engine.
Newton sees:
F ⟶ a.
ECM reveals:
F ⟶ −ΔPEᴇᴄᴍ ⟶ −Mᵃᵖᵖ ⟶ Mᵉᶠᶠ ⟶ a.
This is why ECM can generate gravity, inertia, dark mass, and cosmic dynamics from a single physical process — the manifestation of mass from potential energy.
20. ECM Insight: Mass is Not Static in Motion
In Extended Classical Mechanics (ECM), motion is powered by mass–energy conversion, governed by the manifestation identity
−ΔPEᴇᴄᴍ ↔ ΔKEᴇᴄᴍ ↔ ΔMᴍ
This means that when potential energy is released, it does not disappear into abstract “energy”; it manifests as matter-mass.
Therefore, a moving body possesses more mass than its rest matter mass:
Mᴍₘₒₜᵢₒₙ = Mᴍᵣₑₛₜ + ΔMᴍ
where ΔMᴍ is the kinetic mass created from the released potential energy −ΔPEᴇᴄᴍ.
21. Total Energy in ECM
Classically,
Eₜₒₜₐₗ = PE + KE = mgh + ½mv².
In ECM,
Eᴇᴄᴍ,ₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ
But ECM resolves how this sum is formed:
PEᴇᴄᴍ + KEᴇᴄᴍ = (PEᴇᴄᴍ - ΔPEᴇᴄᴍ) + KEᴇᴄᴍ
Since
KEᴇᴄᴍ ≡ −ΔPEᴇᴄᴍ ≡ −Mᵃᵖᵖ ≡ ΔMᴍ,
we obtain
Eᴇᴄᴍ,ₜₒₜₐₗ = PEᴇᴄᴍᵉᶠᶠ + KEᴇᴄᴍ,
where:
PEᴇᴄᴍᵉᶠᶠ = PEᴇᴄᴍ − ΔPEᴇᴄᴍ = Mᵉᶠᶠgᵉᶠᶠh
This expresses that motion reduces stored potential energy and creates kinetic mass.
22. Potential Energy of the Effective Mass
The released energy modifies the gravitationally active mass.
The system now carries effective mass
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ),
so the effective potential energy becomes
PEᴇᴄᴍᵉᶠᶠ = Mᵉᶠᶠgᵉᶠᶠh
Thus gravity in ECM acts on matter mass plus manifested NAM.
23. Kinetic Energy as Frequency-Generated Mass
In ECM, kinetic energy is not an abstract term — it is mass oscillating at frequency.
The inherent (de Broglie) component is
KEᴇᴄᴍ,ᵢₙₕₑᵣₑₙₜ = ½(−Mᵃᵖᵖ,ᵢₙₕₑᵣₑₙₜ)c² = ½(ΔMᴍ,ᵢₙₕₑᵣₑₙₜ)c² = ½(fᵈᴮ)c².
When this inherent oscillation interacts with the gravitational source field Mɢ, a Planck-type interactional component appears. The total kinetic energy becomes
KEᴇᴄᴍ↑ = ½(−Mᵃᵖᵖ,ᵢₙₕₑᵣₑₙₜ −Mᵃᵖᵖ,ᵢₙₜₑᵣₐᴄₜᵢₒₙₐₗ)c² = ½(fᵈᴮ + fᴾ)c² = ΔMᴍc² = hf
Thus,
KEᴇᴄᴍ = ΔMᴍc² = hf.
For photons (pure oscillatory systems),
KEᴇᴄᴍ = ΔMᴍc² = hf.
24. Frequency-Governed Kinetic Energy Law
KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² = (αΔMᴍᵈᴮ + (1 − α)ΔMᴍᴾ)c² = h(fᵈᴮ + fᴾ) = hf
This shows that
• de Broglie frequency governs inertial motion
• Planck frequency governs field-coupled motion
• their sum governs total kinetic mass and energy
Final ECM Meaning
Classical mechanics writes
KE = ½mv².
ECM reveals the hidden structure:
KEᴇᴄᴍ = ΔMᴍc² = hf,
where the mass ΔMᴍ is created by
−ΔPEᴇᴄᴍ
Motion is therefore the physical manifestation of mass released from potential energy.
This is the engine beneath Newton’s equations.
25. Alphabetical List of ECM Terms and Denotations
α (alpha)
Weighting coefficient that determines how much of the total manifested kinetic mass comes from de Broglie (inertial) versus Planck (field-coupled) frequency contributions.
aᵉᶠᶠ
Effective acceleration produced by the action of force on the effective mass Mᵉᶠᶠ.
c
Speed of light. In ECM it acts as the mass-to-frequency conversion constant via KEᴇᴄᴍ = ΔMᴍc² .
ΔKEᴇᴄᴍ
Change in kinetic energy in ECM, generated by the manifestation of mass from released potential energy.
ΔMᴍ
Manifested matter-mass produced from released potential energy.This is the physical carrier of kinetic energy.
ΔPEᴇᴄᴍ
Change in ECM potential energy. A negative change −ΔPEᴇᴄᴍ generates NAM and kinetic mass.
f
Total effective frequency associated with manifested mass:
f = fᵈᴮ + fᴾ
fᵈᴮ (de Broglie frequency). Frequency associated with inertial motion of manifested mass.
fᴾ (Planck frequency). Frequency associated with field-coupled (gravitational) interaction of manifested mass.
gᵉᶠᶠ
Effective gravitational field acting on the effective mass Mᵉᶠᶠ.
h
Planck constant, relating frequency to manifested energy-mass via
KEᴇᴄᴍ = hf = ΔMᴍc²,
KEᴇᴄᴍ
Kinetic energy in ECM. It is not abstract motion energy but mass created from released potential energy:
KEᴇᴄᴍ = hf = ΔMᴍc²
Mᵃᵖᵖ (Negative Apparent Mass, NAM). Mass equivalent of released potential energy:
−Mᵃᵖᵖ ≡ −ΔPEᴇᴄᴍ
It is the field-like mass that drives gravity, inertia, and kinetic energy.
Mᵉᶠᶠ (Effective Mass)
Total gravitational and inertial mass of a moving system:
Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ).
Mɢ (Gravitational Mass)
In ECM: Mᵉᶠᶠ = Mɢ
Gravity responds to both matter mass and NAM.
Mᴍ (Matter Mass)
The rest-existence mass of matter — the static mass stored in the potential field PEᴇᴄᴍ.
Mᴍ,ₘₒₜᵢₒₙ
Total mass of a moving body:
Mᴍ,ₘₒₜᵢₒₙ = Mᴍ,ᵣₑₛₜ + ΔMᴍ
PEᴇᴄᴍ
Stored ECM potential energy — the existence field carried by matter mass Mᴍ.
PEᴇᴄᴍᵉᶠᶠ
Effective potential energy after manifestation:
PEᴇᴄᴍᵉᶠᶠ = PEᴇᴄᴍ − ΔPEᴇᴄᴍ = Mᵉᶠᶠgᵉᶠᶠh
−ΔPEᴇᴄᴍ
Released ECM potential energy that creates NAM and kinetic mass.
−Mᵃᵖᵖ
Negative Apparent Mass created by −ΔPEᴇᴄᴍ.
It is the engine of gravity, inertia, and kinetic energy.
ΔMᴍᵈᴮ
Portion of manifested mass generated by de Broglie (inertial) oscillations.
ΔMᴍᴾ
Portion of manifested mass generated by Planck (field-coupled) oscillations.
ΔMᴍc²
Energy equivalent of manifested mass:
ΔMᴍc² = hf
Phase-Kernel
The 0-D ECM source where frequency-driven mass manifestation occurs, generating NAM, gravity, and kinetic energy.
NAM (Negative Apparent Mass)
The mass-equivalent of released potential energy.
It produces gravitational attraction, inertial resistance, and kinetic energy.
Weak Equivalence Principle (Classical Limit)
In ECM collapse limit −Mᵃᵖᵖ → 0,
Mᵉᶠᶠ → Mᴍ
so inertial mass equals gravitational mass.
