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

1. Thakur S.N. (2025). A Nuanced Perspective on Dark Energy: Extended Classical Mechanics. Int. J. Astron. Mod. Phys., 01(01), 2025;01(1):001. https://magnivelinternational.com/journal/articledetails/28

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.

Oversimplification of Einstein’s Original 1905 Postulates

May 13, 2025

In a critical and insightful clarification, physicist André Michaud revisits the foundational postulates of Einstein’s 1905 paper Zur Elektrodynamik bewegter Körper (On the Electrodynamics of Moving Bodies), highlighting a significant divergence between Einstein’s original wording and the modern textbook formulations of special relativity.

The Common Modern Interpretation

Today, the postulates of special relativity are typically summarized as:

1. The speed of light in a vacuum is invariant in all inertial frames of reference.

2. The laws of physics are the same in all inertial frames of reference.

While this version has become standard, Michaud points out that this is not how Einstein originally expressed these principles. The current phrasing subtly shifts the focus of Einstein’s arguments and introduces interpretative assumptions that were not explicitly stated in 1905.

Einstein’s Actual Formulations in 1905

First Postulate (1905 Original):

“Sich das Licht im leeren Raume stets mit einer bestimmten, vom Bewegungszustande des emittierenden Körpers unabhängigen Geschwindigkeit V fortplanze.”

Translation: “Light always propagates in empty space at a certain speed V independent of the state of motion of the emitting body.”

This formulation centres on the independence of light’s speed from the motion of its source, rather than asserting its invariance across all observers or frames. The distinction is subtle but significant: Einstein emphasized emission independence, not frame-invariant observation.

Second Postulate (1905 Original):

“Für alle Koordinatensysteme, für welche die mechanischen Gleichungen gelten, auch die gleichen elektrodynamischen und optischen Gesetze gelten.”

Translation: “For all coordinate systems for which the mechanical equations apply, the same electrodynamic and optical laws also apply.”

Here, Einstein asserts the applicability of electromagnetic and optical laws within the same frames that respect Newtonian mechanics—inertial frames. He did not claim universal symmetry of all physical laws across all reference frames, as is often implied in later interpretations.

Historical Context and Neglected Work

Michaud further contextualizes this misinterpretation by drawing attention to a critical moment in physics history. In 1907, the growing acceptance of Special Relativity led the scientific community to set aside the earlier efforts of Wilhelm Wien, who had attempted to synchronize electromagnetic theory with kinetic mechanics. According to Michaud, this promising line of inquiry was prematurely abandoned, leaving a gap in the unified understanding of motion and field dynamics.

Michaud contends that this unification has now been achieved in his work titled:

"Electromagnetic and Kinematic Mechanics Synchronized in Their Common Frame of Reference."

This study seeks to fulfill the original spirit of unification Einstein pursued—through a framework that reintegrates electrodynamics with classical inertial motion principles on more physically grounded terms.

Scientific and Philosophical Implications

The importance of Michaud’s clarification extends beyond historical accuracy. It opens a broader discussion about how foundational postulates are transmitted, reinterpreted, and often oversimplified in the progression of scientific paradigms. By returning to Einstein’s original German text, Michaud demonstrates how nuanced and context-sensitive Einstein’s thinking was, and how easily such nuance can be lost when distilled into modern axioms.

His analysis encourages physicists and theorists to engage more critically with the assumptions embedded in postulates and to re-examine whether alternative or complementary formulations—such as those emerging in Extended Classical Mechanics (ECM) or wave-based dynamics—may offer more complete or realistic descriptions of physical phenomena.