Mass-Energy Dynamics in Extended Classical Mechanics (ECM)

Soumendra Nath Thakur

March 04, 2025

In Classical Mechanics, when kinetic energy is zero (KE=0), the total energy of the system is entirely in the form of potential energy (Eₜₒₜₐₗ = PE), which is associated with the rest mass (m) of the object.

When a force (F) is applied to the object, it accelerates, resulting in an increase in kinetic energy (KE). The total energy of the system is then the sum of potential and kinetic energy:

Eₜₒₜₐₗ = PE + KE

During energy transformation, a portion of the stored energy (PE) is converted into kinetic energy (KE). This transformation can be expressed as:

KE  = ΔPE, so that Eₜₒₜₐₗ = (PE − ΔPE) + ΔPE 

Initially, all of the system's energy is in the form of stored energy (PE). As the system moves, part of this energy is used to generate motion, reducing the stored energy to PE−ΔPE, while the extracted portion becomes kinetic energy (KE=ΔPE).

Despite this redistribution, the total energy remains unchanged; only its allocation between stored energy and motion energy shifts. This balance is maintained by the inverse relationship:

PE ∝ 1/KE = 1/ΔPE

Thus, any reduction in stored energy results in an equal increase in kinetic energy, ensuring conservation within the system.

Potential Energy, Kinetic Energy, and Mass Relation in ECM:

In ECM, an object's energy is dynamically linked to its motion and gravitational interactions. The relationship between potential energy (PE), kinetic energy (KE), and mass follows an inverse relation, where:

PE ∝ 1/KE = 1/ΔPE

Total Energy at Rest and in Motion:

At rest, the total energy of an object with mass m is entirely in the form of potential energy:

Eₜₒₜₐₗ = PE

When a force is applied, the object undergoes acceleration, leading to a conversion of stored potential energy (PE) into kinetic energy (KE). The total energy expression becomes:

Eₜₒₜₐₗ = PE + KE = (PE − ΔPE) + (ΔPE) 

where ΔPE is the portion of potential energy converted into kinetic energy.

Dynamic Mass Response and Force Relation:

Applying Newton’s second law (F=ma):

Since acceleration is inversely proportional to mass (a∝1/m), increasing acceleration leads to an apparent reduction in effective mass.

This means that as the system gains kinetic energy (KE=ΔPE), the object’s potential energy decreases (PE−ΔPE), and the apparent mass contribution emerges

Apparent Mass and Effective Mass in ECM:

Since kinetic energy is dynamically linked to mass, the corresponding mass equivalent of KE is negative apparent mass:

KE = ∣ΔPE∣ corresponds −Mᵃᵖᵖ = ∣ΔPE∣ 

Since apparent mass is inherently negative, the formulation remains valid without further sign corrections.

Thus, the ECM mass-energy relation is given by:

Eₜₒₜₐₗ,ᴇᴄᴍ = (Mᴍ − Mᵃᵖᵖ) + (−Mᵃᵖᵖ)

which simplifies to:

Eₜₒₜₐₗ,ᴇᴄᴍ = ±Mᵉᶠᶠ + (−Mᵃᵖᵖ)

where:

• Mᴍ is the matter mass,

• −Mᵃᵖᵖ represents the negative apparent mass contribution arising from kinetic energy,

• Mᵉᶠᶠ represents the effective mass, which adjusts dynamically with motion.

Key Interpretation in ECM:

•  Mass is not an intrinsic property but a dynamic response to motion and gravitational interactions.

•  Acceleration reduces the contribution of effective mass, increasing kinetic energy and manifesting as negative apparent mass.

•  The total energy balance remains consistent, with kinetic energy linked to an inverse mass-energy relation.