Restoring Dynamic Mass in Classical Mechanics - The Foundation of Extended Classical Mechanics (ECM)

Chapter: 

DOI: http://doi.org/10.13140/RG.2.2.35190.72009

Soumendra Nath Thakur

• ORCiD: 0000-0003-1871-7803
• Tagore's Electronic Lab, WB, India; postmasterenator@gmail.com or postmasterenator@telitnetwork.in

May 13, 2025

Abstract

This chapter establishes the foundation of Extended Classical Mechanics (ECM) by addressing a fundamental oversight in classical mechanics: the failure to recognize the dynamic mass component inherent in kinetic energy. While classical mechanics accepts the inverse mass relationship a ∝ 1/m in Newton's second law; it never assigns physical meaning to the inverse-mass term. ECM resolves this by interpreting 1/m as a representation of negative apparent mass −Δm, revealing that kinetic energy is carried by a real but negative dynamic mass component.

Building on this insight, ECM reformulates the force equation as:

Fᴇᴄᴍ = (M + (−Mᵃᵖᵖ))⋅aᵉᶠᶠ  = Mᵉᶠᶠ⋅aᵉᶠᶠ

And for systems consisting purely of dynamic energy (like photons), as:

Fᴇᴄᴍ = (−2Mᵃᵖᵖ))⋅aᵉᶠᶠ

This generalized formulation accounts for both static matter and massless energy carriers by incorporating gravitational modulation of dynamic mass. The chapter demonstrates how this correction not only completes and extends classical mechanics, but also offers a consistent mechanical explanation for photon behaviour, repulsive gravitational effects (as in dark energy), and the cosmological gravitating mass Mɢ ​. ECM thus provides a unified framework linking classical force laws with modern and cosmological physics by restoring the missing dynamic mass component to the mechanics of energy.

1. Classical Mechanics and the Hidden Mass of Kinetic Energy

In classical mechanics, the force equation F=ma implies that acceleration is inversely proportional to mass:

a ∝ 1/m

This inverse dependence is mathematically consistent, but its physical meaning is overlooked. If we reinterpret:

1/m = −Δm

We treat the inverse-mass effect as a negative mass component −Δm, linked to the dynamic aspect of energy. Though not recognized classically, this idea introduces a mass contribution originating not from rest matter, but from motions itself—the dynamic mass of kinetic energy.

In the classical energy equation:

Eₜₒₜₐₗ = PE + KE = mgh + ½mv²

We now reinterpret:

• m −Δm as the effective mass for potential energy,

• −Δm as the effective energetic mass contributing to kinetic energy.

Corrected expressions become:

PE = (m − Δm)gh, KE = ½(m−Δm)v² ⇒ −Δm = dynamic energy carrier.

2. The ECM Force Equation and the Role of Negative Apparent Mass

This insight leads directly to the ECM force equation, which accounts for both matter mass Mᴍ​ and negative apparent mass −Mᵃᵖᵖ​:

Fᴇᴄᴍ = (M + (−Mᵃᵖᵖ))⋅aᵉᶠᶠ = Mᵉᶠᶠ⋅aᵉᶠᶠ ​

Where:

• Mᵉᶠᶠ​ is the total effective mass participating in the force interaction,

• aᵉᶠᶠ is the effective acceleration derived from dynamic energy behaviour.

For systems made entirely of dynamic energy (e.g., photons or unbound electromagnetic energy), the ECM force law becomes:

Fᴇᴄᴍ = (−2Mᵃᵖᵖ))⋅aᵉᶠᶠ  

The factor −2 arises from:

• the intrinsic apparent mass −Mᵃᵖᵖ​,

• an additional gravitational modulation analogous to the gravitational energy contribution in massive systems.

3. Extension to Modern and Cosmological Physics

Photon Dynamics

Photons, though massless in conventional models, carry energy and momentum. ECM reconciles this by assigning photons a total negative dynamic mass of −Mᵃᵖᵖ, leading to the photon-specific force law:

Fᴇᴄᴍ = (−2Mᵃᵖᵖ))⋅aᵉᶠᶠ  

This resolves the paradox of force and inertia in light: energy can exert force through effective dynamic mass even in the absence of rest mass.

Dark Energy Analogy

ECM identifies dark energy as a large-scale manifestation of photon dynamic energy characterized by a single negative apparent mass −Mᵃᵖᵖ, not −2Mᵃᵖᵖ. This value represents the inherent energy of a photon after escaping gravitational influence and expending the corresponding potential energy. The remaining −Mᵃᵖᵖ​​ persists in zero-gravity spheres and dark-energy-dominated space and is permanently expended as the photon undergoes cosmic redshift.

Thus, the repulsive cosmological effect attributed to dark energy corresponds to:

Fᴇᴄᴍ = (−Mᵃᵖᵖ))⋅aᵉᶠᶠ

This formulation reveals that the accelerated expansion of the universe is a mechanical consequence of effective force acting on residual dynamic mass in low-gravity space.

Cosmological Models

In cosmology, the gravitating mass Mɢ governs cosmic expansion. ECM reformulates it as:

            Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) = Mɢ

This corresponds directly to the framework in Chernin et al. (2013), where:

• Mᴍ​: matter mass (ordinary + dark matter),

• −Mᵃᵖᵖ: ECM’s analogue of the cosmological dark energy term.

Thus, ECM reinterprets cosmological gravitating mass Mɢ as a special case of its generalized effective mass framework.

4. Conclusion

The classical failure to recognize the negative dynamic mass of kinetic energy created a critical theoretical gap. ECM closes this by:

• assigning real mass to dynamic energy,

• completing classical force and energy laws,

• Providing a consistent mechanical foundation for photons, dark energy, and cosmic expansion.

ECM not only redefines classical mechanics but also establishes a coherent and unified view of modern and cosmological phenomena through the physical reality of dynamic mass.

Appendix: Denotations (Alphabetical List)

Symbol	Description
aᵉᶠᶠ	Effective acceleration
Eₜₒₜₐₗ	Total mechanical energy
F	Classical force
Fᴇᴄᴍ​	Effective force in ECM
g	Gravitational field strength
h	Height in potential energy
KE	Kinetic energy
Mᵃᵖᵖ	Apparent (negative) mass in ECM
​Mᵉᶠᶠ	Effective mass = Mᴍ + (−Mᵃᵖᵖ)
Mɢ​	Gravitating mass in cosmology
​Mᴍ	Matter mass (includes ordinary and dark matter)
PE	Potential energy
v	Velocity
Δm	Equivalent negative dynamic mass from 1/m relationship

Funding

• No specific funding was received for this work.

Conflicts of Interest

• No potential competing interests to declare.

5. References

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2. Thakur, S. N. (2025). Foundational Formulation of Extended Classical Mechanics: From Classical Force Laws to Relativistic Dynamics. Preprints. https://doi.org/10.20944/preprints202504.1501.v1

3. Thakur, S. N., & Bhattacharjee, D. (2023b, October 30). Phase Shift and Infinitesimal Wave Energy Loss Equations. Longdom. https://www.longdom.org/open-access/phase-shift-and-infinitesimal-wave-energy-loss-equations-104719.html

4. Thakur, S. N. (2024). Extended Classical Mechanics: Vol-1 - Equivalence Principle, Mass and Gravitational Dynamics. Preprints. https://doi.org/10.20944/preprints202409.1190.v3

5. Thakur, S. N. & Tagore’s Electronic Lab. (2025). Rotational phase shift and time distortion in a rapidly rotating piezoelectric system. In Tagore’s Electronic Lab [Journal-article]. https://doi.org/10.13140/RG.2.2.24780.32640

6. Thakur, S. N. (2025). Mathematical Derivation of Frequency Shift and Phase Transition in Extended Classical Mechanics (ECM). ResearchGate, 390208822. https://doi.org/10.13140/RG.2.2.36663.02721.