Submitted:
07 August 2025
Posted:
14 August 2025
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Abstract
This paper presents a revised formulation of kinetic energy within Extended Classical Mechanics (ECM), interpreting it as a frequency-governed process arising from mass displacement transitions. ECM proposes that kinetic energy emerges from the redistribution of rest mass (Mᴍ) into a dynamic component (ΔMᴍ), structured by two distinct frequency domains: the de Broglie frequency governing translational motion and the Planck frequency reflecting intrinsic quantum excitation. The resulting kinetic energy relation, KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² = hf, yields the classical ½mv² limit under low-frequency conditions while providing explanatory power for quantum and high-energy phenomena. Applications to atomic transitions, thermionic emission, nuclear fission, and fusion show that observed energy release can be interpreted as frequency-driven mass redistribution rather than annihilation. ECM thus reframes kinetic energy as an emergent property of dual-frequency mass dynamics, offering a unified theoretical lens spanning classical, quantum, and nuclear regimes.
Keywords:
extended classical mechanics (ECM)
; kinetic energy
; frequency-governed energy
; mass displacement (ΔMᴍ)
; de broglie frequency
; planck frequency
; mass-energy transformation
; nuclear fission
; nuclear fusion
; rest-mass energy
; dynamic mass-energy
; hf energy relation
; residual mass redistribution
Introduction
Traditional formulations of mass-energy equivalence [3] do not fully capture the kinetic transformation of dynamic mass-energy. They typically account for the conversion of a portion of rest mass into rest energy but overlook the frequency-governed redistribution that characterizes kinetic transitions. In nuclear reactions, a significant portion of the rest mass remains unconverted and continues to exist as massive particles, even as nuclei undergo processes of splitting (fission) or combining (fusion).
In nuclear fission, a heavy nucleus splits into two or more lighter nuclei, releasing energy and emitting alpha particles, beta particles, and gamma rays during the decay process. In contrast, nuclear fusion involves the combination of lighter nuclei—such as isotopes of hydrogen (deuterium and tritium)—to form a heavier nucleus like helium. This fusion process releases energy primarily in the form of gamma rays and emits alpha particles and neutrons, but notably does not emit beta particles.
Both fission and fusion involve not only a mass-to-energy transformation, but also the retention and redistribution of residual mass. In fission, a portion of the rest mass is transformed into energy, while the residual mass remains distributed among the lighter daughter nuclei and the emitted particles [7,8]. In fusion, a small portion of the combined lighter nuclei’s mass becomes mass-energy, while the rest is retained in the newly formed heavier nucleus and associated particles.
Beyond nuclear reactions, the ECM kinetic energy formulation also applies to atomic-scale transitions. In particular, photon emission during electronic orbital changes—whether involving relaxation to lower energy levels or ionization—can be interpreted as frequency-governed rearrangements of ΔMᴍ [i,x]. These emissions reflect partial releases of kinetic energy governed by the superposition of intrinsic (Planck) and translational (de Broglie) mass-frequency domains. This reinforces the universality of the ECM framework across quantum, atomic, and nuclear regimes [9] [iv–vii].
While this paper focuses on theoretical development, it does not attempt direct empirical data analysis or measurement [x].
Accordingly, to address the foundational question—Why combine both ΔMᴍᵈᴮ and ΔMᴍᴾ in the formulation of the frequency-driven kinetic energy equation in ECM?—we shall first outline the conceptual methodology, followed by its corresponding mathematical formulation [x]:
Methodology
The methodological foundation of this study involves a comparative and reconstructive analysis of kinetic energy within the framework of Extended Classical Mechanics (ECM), guided by dual-frequency contributions arising from de Broglie and Planck regimes. Rather than treating kinetic energy solely as a function of velocity and inertial mass—as in classical or relativistic formulations—ECM reinterprets it as a frequency-governed manifestation of dynamic mass displacement, ΔMᴍ, as a redistribution of dynamic mass through frequency excitation [v], [vi],[9].
The approach proceeds through the following conceptual stages:
1. Critical Assessment of Relativistic Kinetics:
2. Reconstruction of Kinetic Energy via ΔMᴍ:
ECM introduces the concept of dynamic mass displacement (ΔMᴍ) as the primary carrier of kinetic energy [i], [ii], [v], and framing motion not in terms of mere velocity but through transitions in internal mass structure governed by frequency.
3. Dual-Frequency Integration:
A dual-mode interpretation is applied, combining:
- ΔMᴍᴾ: representing microscopic intrinsic (Planck) dynamics [6], [iv].
These are superposed to yield a composite displacement:
ΔMᴍ = ΔMᴍᵈᴮ + ΔMᴍᴾ, [vi].
4. Nonlinear Frequency-Governed Expression:
The kinetic energy is thus expressed as:
KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² = hf, [vi]
where f = fᵈᴮ + fᴾ denotes the total effective frequency.
5. Empirical Illustration Through Nuclear Reactions:
6. Restoration of Classical Principle:
The ECM formulation asymptotically converges to the classical ½mv² regime at low-frequency (macroscopic) limits [5], [ii], while revealing its broader application across quantum and relativistic domains.
Mathematical Presentation
1. Primary Total Energy Relation in ECM
ECM defines total energy as a mass-based redistribution between potential and kinetic contributions:
Eₜₒₜₐₗ = PEᴇᴄᴍ + KEᴇᴄᴍ ⇒ (Mᴍ− ΔMᴍ) + ΔMᴍ [i]
where:
- Mᴍ: Total matter mass
- ΔMᴍ: Displaced mass component associated with kinetic excitation
- −ΔMᴍ = −Mᵃᵖᵖ: Apparent mass experienced in dynamic states
- ΔMᴍ ⇒ KEᴇᴄᴍ: Represents the mass fraction converted to kinetic energy
In this formalism, ECM frames energy as a realignment of mass components:
A retained mass contributes to potential energy (e.g., gravitational), while the displaced mass governs kinetic energy, emerging through motion- or frequency-induced redistribution.
2. Full Energy Equation in ECM
ECM incorporates gravitational effects and kinetic motion via an effective-mass-based energy equation:
Eₜₒₜₐₗ = Mᵉᶠᶠ gᵉᶠᶠ h + ½Mᵉᶠᶠ v² [i], [iii]
where:
- Mᵉᶠᶠ = Mᴍ − ΔMᴍ: Effective mass under gravitational influence
- gᵉᶠᶠ: Local effective gravitational field strength
- v: Velocity of the particle or system
This expression reinforces ECM’s central view: energy is not added as an external abstraction but arises through redistribution of actual mass—potential energy is derived from the undisturbed mass (Mᵉᶠᶠ), and kinetic energy from the ΔMᴍ displaced during motion.
3. ECM Kinetic Energy at the Photon Limit
For a massless particle (e.g., photon) at light speed (v = c), ECM reformulates kinetic energy as:
KEᴇᴄᴍ = ½Mᵉᶠᶠc² = hf [v], [iv]
Given the following identity for photon-like dynamics:
Mᵉᶠᶠ = −Mᵃᵖᵖ−Mᵃᵖᵖ = −2Mᵃᵖᵖ,
We derive:
½(−2Mᵃᵖᵖ)c² = −Mᵃᵖᵖc² = hf
This also aligns with:
KEᴇᴄᴍ = ΔMᴍc² = hf
Thus, kinetic energy of a photon emerges solely from mass displacement (ΔMᴍ)—not from rest mass. The equivalence hf = ΔMᴍc² highlights that photon energy is a frequency-induced manifestation of dynamic mass within ECM, without invoking special relativity.
4. Why Combine Both ΔMᴍᵈᴮ and ΔMᴍᴾ?
ECM reconceptualise kinetic energy not as a scalar tied to velocity, but as a frequency-governed dynamic mass displacement (ΔMᴍ), incorporating two frequency regimes [1,2,6], [iv]:
- ΔMᴍᵈᴮ: Represents translational motion within the de Broglie domain (macroscopic scale; λ → ∞)
- ΔMᴍᴾ: Captures intrinsic quantum excitation in the Planck domain (microscopic scale; λ → 0)
These waveforms are not mutually exclusive. Rather, they are complementary, and must be superposed to fully account for the emergence of kinetic energy in ECM.
5. Unified ECM Kinetic Energy Expression
KEᴇᴄᴍ = [ΔMᴍ⁽ᵈᴮ⁾ + ΔMᴍ⁽ᴾ⁾]c² = ΔMᴍc² = hf
- hf is not symbolic—it reflects the total effective frequency governing ΔMᴍ
- The expression is nonlinear and frequency-dominant [vi], [ix], replacing both Newtonian (½mv²) and relativistic (γmc² − mc²) forms
- At low velocities, the Planck contribution (ΔMᴍᴾ) dominates [6].
6. Frequency Composition and Energy Redistribution
From ECM’s viewpoint, ΔMᴍ represents redistributed mass-energy, not annihilated mass:
Eₜₒₜₐₗ = Mᵉᶠᶠgᵉᶠᶠh + ½ΔMᴍv²
At transition limits (v → c):
ΔMᴍc² = KEᴇᴄᴍ = hf,
where:
f = f⁽ᵈᴮ⁾ + f⁽ᴾ⁾, [9][vi,vii]
This shows that kinetic energy is governed by the superposition of de Broglie and Planck frequencies.
7. Application to Nuclear Reactions
ECM interprets fission and fusion not as total mass-to-energy conversions, but as redistributions of mass to frequency-governed mass-energy [7,8], [ix]:
(Mᴍ− ΔMᴍ) + ΔMᴍ,
where:
ΔMᴍc² = KEᴇᴄᴍ = hf
This mass displacement gives rise to observable energetic emissions: gamma rays, alpha and beta particles, and neutrons.
8. Justification for Dual Frequency Limits
The need to incorporate both frequency domains arises from empirical and conceptual necessity:
- The de Broglie limit (λ → ∞) governs large-scale translational motion
-
The Planck limit (λ → 0) governs internal quantum excitationsTogether, they define a complete frequency spectrum of dynamic mass transitions. Hence,
KEᴇᴄᴍ = [ΔMᴍ⁽ᵈᴮ⁾ + ΔMᴍ⁽ᴾ⁾]c² = ΔMᴍc² = hf, [6], [iv], [ix]
This synthesis reveals that kinetic energy is not a fixed scalar, but a dynamically emergent phenomenon—rooted in the frequency-governed restructuring of mass across both macroscopic and quantum regimes, reflects the internal mass-energy distribution across both domains [iv], [vi],[9], [vii].
Discussion
The kinetic energy formulation in Extended Classical Mechanics (ECM), expressed as:
KEᴇᴄᴍ = [ΔMᴍ⁽ᵈᴮ⁾ + ΔMᴍ⁽ᴾ⁾]c² = hf,
represents a significant departure from traditional Newtonian or relativistic frameworks. In conventional physics, kinetic energy is typically framed either as a function of velocity (½mv²) or as a relativistic correction (γmc² − mc²). These expressions, while successful within their respective domains, are scalar formulations that lack integration with the frequency properties of mass-energy transformations [3,4,5].
In contrast, ECM posits that mass undergoing motion (Mᴍ) undergoes a mass-displacement (ΔMᴍ) governed by frequency-domain transitions. The kinetic energy emerges not from mere spatial translation but from a redistribution of mass through dual-frequency excitations [vi], [x]:
- One associated with de Broglie frequencies (fᵈᴮ) due to macroscopic translational motion.
- The other with Planck-scale frequencies (fᴾ) associated with microscopic, intrinsic excitations.
The coherence and necessity of combining these two components are empirically motivated.
For example, in nuclear reactions (such as fission or fusion), the energy released is traditionally attributed to a change in mass (Δm) via Einstein’s identity E = Δmc². However, this scalar view cannot fully explain the structured frequency emission patterns—such as gamma-ray spectra—or the particulate by products (e.g., alpha, beta, and neutron emissions) observed across reactions [7,8], [ix].
By incorporating ΔMᴍ⁽ᵈᴮ⁾ and ΔMᴍ⁽ᴾ⁾, ECM explains such nuclear phenomena as frequency-governed transitions of the dynamic mass state. The dual-frequency formulation explains why both macroscopic momentum changes and microscopic quantum excitations contribute simultaneously to the observed kinetic energy release. This implies that kinetic energy is not simply “gained” by a particle—but is emergent from internal restructuring of mass-energy, modulated by a total frequency f = fᵈᴮ + fᴾ.
Moreover, ECM’s formulation brings renewed clarity to the luminal limit (v → c). At such thresholds, translational kinetic energy saturates, yet frequency-governed transformations may still proceed, particularly via ΔMᴍᴾ, allowing ECM to remain analytically valid even where classical or relativistic expressions fail. This supports the idea that mass-energy transformations near the speed of light involve intrinsic excitation of mass, rather than merely asymptotic velocity increases.
The implications extend beyond high-energy physics. Any process involving particle motion—be it thermionic emission, Compton scattering, or pair production—may be reinterpreted through the lens of ΔMᴍ-frequency redistribution, allowing kinetic energy to be visualized as a spectral property rather than a scalar quantity [x], [iv].
Finally, the ECM interpretation challenges the idea that mass is destroyed or converted during energetic processes. Instead, ECM proposes a redistribution scheme [7], [viii]:
(Mᴍ− ΔMᴍ) + ΔMᴍ,
where ΔMᴍc² = hf. The first term corresponds to the residual dynamic mass, while the second encodes the frequency-governed energy displacement that manifests observably.
In sum, ECM’s kinetic energy formulation, grounded in dual-frequency mass displacement, provides a richer, multi-domain understanding of mass-energy transitions. It offers a unified model applicable to microscopic quantum events and macroscopic inertial motion, both, revealing kinetic energy as a composite frequency response rather than a fixed classical quantity.
Conclusions
The formulation of kinetic energy within Extended Classical Mechanics (ECM) as:
KEᴇᴄᴍ = [Mᴍᵈᵉᴮʳᵒᵍˡᶦᵉ + ΔMᴍᴾˡᵃⁿᶜᵏ] c² = ΔMᴍc² = hf, [vi]
introduces a fundamentally new understanding of motion-induced energy transformation. Unlike conventional models where kinetic energy is derived from velocity or inertial mass alone, ECM reveals that kinetic energy is a mass-to-mass-energy transition governed by dual-frequency contributions:
- A de Broglie component (ΔMᴍ⁽ᵈᵉᴮ⁾) reflecting macroscopic translational motion [iv], [vi].
- A Planck component (ΔMᴍ⁽ᴾ⁾) reflecting microscopic quantum excitation [iv], [vi].
By combining these two domains, ECM delivers a coherent and unified description of kinetic energy that remains valid across both classical and quantum regimes.
This dual-frequency basis not only refines the mass-energy relation but also provides explanatory power for processes where conventional theories face limitations—such as in thermionic emission, nuclear decay, or high-velocity transitions near luminal speeds [6,8], [ix], [x].
Importantly, the ECM approach reframes the view of mass–energy interactions. Rather than treating energy emission as mere mass loss, it emphasizes a redistribution of dynamic mass through frequency excitation [v], [vi],[9]. The resulting kinetic energy is not just a byproduct of motion but an intrinsic expression of the system’s composite frequency state.
Therefore, this framework allows kinetic energy to be interpreted as the observable outcome of internal frequency restructuring within matter—a restructuring that respects both de Broglie and Planck regimes.
This perspective not only advances the theoretical landscape of classical mechanics but also provides a versatile bridge to quantum and relativistic interpretations [3,4,6].
Funding: No financial aid is received for this work.
Conflicts of Interest: Author declares no conflict of Interest.
Relevant ECM Appendices
The following ECM appendices provide foundational, methodological, and empirical support for the frequency-based interpretation of kinetic energy and the dual contributions from de Broglie and Planck domains in Extended Classical Mechanics (ECM):
- Appendix 3: Fundamental Total Energy in Extended Classical Mechanics (ECM). https://doi.org/10.13140/RG.2.2.21532.19841 Establishes the total energy formulation in ECM, including the mass-energy partitioning involving rest and dynamic components.
- Appendix 5: ECM Perspective on Classical Kinetic Energy and Mass Displacement. https://doi.org/10.13140/RG.2.2.15962.64962. Presents the transition from classical ½mv² to frequency-governed kinetic energy via ΔMᴍ transitions.
- Appendix 13: Relativistic Interactions and ECM Formalism https://doi.org/10.13140/RG.2.2.33885.46562. Provides ECM-compatible corrections to relativistic kinetic energy, supporting a frequency-based reinterpretation of mass energy dynamics.
- Appendix 24: Wave-Particle Duality and de Broglie–Planck Domains in ECM. https://doi.org/10.13140/RG.2.2.30733.90082. Justifies the inclusion of both de Broglie and Planck frequencies in the kinetic energy framework.
- Appendix 29: Mass Displacement ΔMᴍ and its Frequency Equivalence in ECM. https://doi.org/10.13140/RG.2.2.25010.85447. Analyzes ΔMᴍ as the core transitional construct in ECM kinetic processes, bridging internal and observable motion.
- Appendix 30: Unifying Kinetic Energy through Frequency: ECM Interpretations. https://doi.org/10.13140/RG.2.2.31737.62561. Derives the expression: KEᴇᴄᴍ = (ΔMᴍᵈᴮ + ΔMᴍᴾ)c² and explores its applications.
- Appendix 32: Energy Density Structures in Extended Classical Mechanics (ECM). https://doi.org/10.13140/RG.2.2.22849.88168. Explores time and frequency as conjugates, essential for interpreting kinetic energy via wave dynamics.
- Appendix 35: Mass-Energy Conservation through ΔMᴍ in ECM. https://doi.org/10.13140/RG.2.2.11643.00808. Discusses mass-energy redistribution rather than annihilation, supporting ECM’s interpretation of fission and fusion.
- Appendix 37: Consistent Frequency–Energy–Radius Dynamics in ECM. https://doi.org/10.13140/RG.2.2.21834.07362. Confirms radial and frequency dependencies of ΔMᴍ transformations in bounded and free systems.
- Appendix 40: Empirical Support for ECM Frequency-Governed Kinetic Energy via Thermionic Emission in CRT Systems. https://doi.org/10.13140/RG.2.2.31184.42247
Provides experimental grounding for the frequency-based ECM kinetic energy using CRT thermionic emission as a test case.
Alphabetical List of Relevant Mathematical Terms:
- c – Speed of light in vacuum. Serves as the conversion constant between mass and energy in relativistic and ECM formulations.
- Δm – Infinitesimal mass displacement or loss, often used in classical approximations or radiation interactions; distinguished from ECM’s ΔMᴍ.
- ΔMᴍ – Mass displacement in ECM representing the transition of rest mass into dynamic (kinetic) forms; decomposed into frequency equivalents.
- ΔMᴍ⁽ᵈᴮ⁾ – That portion of ΔMᴍ attributed to the de Broglie frequency of a particle, encoding its observable wave–momentum characteristics.
- ΔMᴍ⁽ᴾ⁾ – That portion of ΔMᴍ attributed to the Planck frequency, representing internal or high-energy clock-like oscillations of the mass-energy system.
- Eₜₒₜₐₗ – Total energy in ECM, often expressed as Mᵉᶠᶠc², and inclusive of both rest and frequency-based kinetic energy components.
- f – Frequency; used generally in expressions like hf, where frequency is associated with a wave or transition state of mass-energy.
- Fᴇᴄᴍ – Force expression in Extended Classical Mechanics, typically incorporating frequency and mass-displacement dependencies.
- gᵉᶠᶠ – Effective acceleration as interpreted in ECM, factoring both observable and internal dynamic effects.
- hf – Planck relation for energy (E = hf); reinterpreted in ECM as representing energy of a transitional mass component ΔMᴍ.
- KEᴇᴄᴍ – Kinetic energy in Extended Classical Mechanics, derived from ΔMᴍ transitions and expressed as: KEᴇᴄᴍ = (ΔMᴍ⁽ᵈᴮ⁾ + ΔMᴍ⁽ᴾ⁾)c² = ΔMᴍc² = hf
- Mᵉᶠᶠ – Effective mass under ECM interpretation; represents the total mass content including dynamic and static components.
- Mᴍ – Mechanical mass in ECM, i.e., the rest-bound mass that may partially convert into kinetic forms under motion or interaction.
- Mᴍ,ᴋᴇ – That part of mechanical mass specifically associated with kinetic energy contributions (linked to ΔMᴍ).
- Mᴍ,ʀₑₛₜ – Residual mechanical mass not involved in kinetic transition; i.e., Mᴍ − ΔMᴍ.
- −ΔMᴍ – Notationally emphasizes mass lost (or converted) from mechanical mass during transition to kinetic or radiative form.
- −Mᵃᵖᵖ – Apparent loss of rest mass as perceived externally in kinetic processes; the observable projection of ΔMᴍ.
- −2Mᵃᵖᵖ – Doubling of apparent mass conversion due to symmetrical or conjugate transitions, often used in bound particle systems or collisions.
- v – Classical velocity of a particle; retained in ECM only as a secondary variable subordinate to frequency-based mass transition terms.
- ½Mᵉᶠᶠv² – Classical kinetic energy expression, maintained for comparison but reinterpreted in ECM as approximating a frequency-governed transition.
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