ECM - Two-Body & Multi-Body Stability (Sections 1-5)

Abstract

This manuscript establishes the foundational architecture of two-body and multi-body stability within the Extended Classical Mechanics (ECM) framework. By replacing curvature-dependent interpretations with ECM's mass-exchange formalism, the analysis demonstrates that gravitational and anti-gravitational behaviour arises from the sign and evolution of effective mass \(M^{\text{eff}}\). Sections 1-5 derive stable, unstable, and transitional regions for positive and negative effective-mass domains, constructing a generalizable ECM phase-space for realistic astrophysical and cosmological systems.

1. Balance Gravitational Point Between Two Bodies (ECM Interpretation)

In Extended Classical Mechanics (ECM), gravitational balance between two bodies arises not from equality of classical forces but from equality of mass-exchange intensities. The defining condition is:

\( \big|\Delta M_{M,1}(r)\big| = \big|\Delta M_{M,2}(r)\big| \)

where \( \Delta M_{M,1} \equiv -M^{\text{app}}_1 = \Delta PE^{\text{ECM}}_1 \) and \( \Delta M_{M,2} \equiv -M^{\text{app}}_2 = \Delta PE^{\text{ECM}}_2 \). This ensures the expression is interpreted strictly in the ECM framework.

Each body possesses an effective mass:

\( M^{\text{eff}} = M_M - M^{\text{app}} \)

The ECM gravitational field is expressed as

\( g^{\text{eff}}(r) = -\dfrac{\Delta M_M(r)}{r^2} \)

So the balance point between two bodies separated by distance d satisfies

\( \dfrac{M_1^{\text{eff}}}{r^2} = \dfrac{M_2^{\text{eff}}}{(d-r)^2} \)

With the algebraic solution for the equilibrium location:

\( r = \dfrac{d}{1 + \sqrt{M_2^{\text{eff}} / M_1^{\text{eff}}}} \)

This formulation emphasizes that ECM balance is a mass-exchange equilibrium, not force equality or spacetime curvature. For foundational definitions of effective and apparent mass, see [Appendix A] and the treatment of negative apparent mass in [Appendix D].

2. Balance Between Gravitational and Anti-Gravitational (\(M^{\mathrm{eff}} < 0\)) Fields

ECM accommodates repulsion when

\( M^{\text{app}} > M_M \quad\Rightarrow\quad M^{\text{eff}} < 0 \)

The balance condition between a positive M^eff (attractive) and a negative M^eff (repulsive) source reads:

\( \left|\dfrac{M_{+}^{\text{eff}}}{r^2}\right| = \left|\dfrac{M_{-}^{\text{eff}}}{(d-r)^2}\right|

At the interface, the net mass-exchange vanishes:

\( \Delta M_{M,(+)}(r) + \Delta M_{M,(-)}(r) = 0 \)

In ECM this anti-gravity mechanism is described in detail in [Appendix D], and related mass-polarity discussion is in [Appendix 33] and [Appendix 34].

3. Radial Instability and Transition Zones Around Negative Effective Mass Regions

Regions where M^eff changes sign produce instability shells defined by:

\( M^{\text{eff}}(r) = 0 \)

This surface marks the threshold where ΔMᴍ switches sign and trajectories shift between bound (inward) and expelled (outward) behaviour. A finite transition zone surrounds this boundary where both contributions coexist and small perturbations decide the outcome.

Mathematically, the total gradient may be written as:

\( \Delta M_{M}^{\text{(tot)}}(r) \propto \dfrac{M_{+}^{\text{eff}} + M_{-}^{\text{eff}}}{r^2} \)

Important references on mass redistribution and internal force generation that underpin instability analysis: [Appendix 11], [Appendix 17], and the frequency-radius dynamics relevant for gradient scaling [Appendix 37].

4. Dynamic Evolution of `M^eff` in Multi-Body Systems

Effective mass is time-dependent:

\( M^{\text{eff}}(r,t)=M_M(t)-M^{\text{app}}(r,t) \)

Sources of evolution include:

Kinetic-to-Mass transitions, \( \Delta M_M = \mathrm{KE}/c^2 \)
Gravitational mass exchange interactions, \( \Delta M_M^{(G)} \)
Redistribution of apparent mass \( M^{\text{app}} \) due to surrounding bodies
Local amplification driving \( M^{\text{app}} > M_M \)

For N bodies the net mass-exchange gradient acting on body i is:

\( \Delta M_{M,i}^{\mathrm{(tot)}}(t)=\sum_{j\neq i}\dfrac{M_j^{\mathrm{eff}}(t)}{r_{ij}(t)^2} \)

See discussions of effective acceleration and mediation in multi-body reversible dynamics: [Appendix 12], and variable matter-mass considerations: [Appendix 47]. Frequency-based contributions to mass evolution are addressed in [Appendix 23 - Part-2] and [Appendix 37].

5. Stability Maps and ECM Phase Surfaces in Multi-Body Fields

Stability is defined by overlapping effective-mass fields:

\( M^{\text{eff}}(r)=M_{(+)}^{\text{eff}}(r)+M_{(-)}^{\text{eff}}(r) \)

Classify space by the sign of the summed ECM field:

\( \Phi_{\text{ECM}}(r)=\sum_i\Delta M_{M,i}(r) \)

Then:

\( \Phi < 0 \) - attractive gravitational basin
\( \Phi > 0 \) - repulsive cavity (anti-gravity)
\( \Phi = 0 \) - ECM phase surface (neutral equilibrium)

Phase surfaces generalize classical equipotentials but are defined by mass-exchange parity rather than potential energy. The energy-density and partitioning frameworks underpinning these surfaces are in [Appendix 32], [Appendix 33], and [Appendix 34].

Note: All notation and sign conventions adhere to ECM conventions: M^eff (effective mass; gravitationally active), < code>M^app (apparent mass), ΔM_M (mass-exchange), and g^eff (effective mass-exchange field). No curvature-based language or cosmological constant is used.

Conclusion

This document establishes ECM foundation for gravitational and anti-gravitational stability structures across two-body and multi-body configurations. The ECM mass-exchange paradigm replaces curvature-based interpretations, enabling a unified framework where stability, repulsion, equilibrium, and phase surfaces emerge directly from the behaviour of \(M^{\text{eff}}\). This forms the basis for further cosmological, astrophysical, and quantum-scale ECM extensions.

References

The following ECM Appendices and Supplements provide the foundational literature for the concepts discussed in Sections 1-5:

Appendix A - Standard Mass Definitions in ECM. DOI: https://doi.org/10.13140/RG.2.2.31762.36800
Appendix D - Negative Apparent Mass and Mass Continuity in ECM. DOI: https://doi.org/10.13140/RG.2.2.10264.92165
Appendix 11 - Mass Redistribution and the Fourfold Structure of Mass in ECM. DOI: https://doi.org/10.13140/RG.2.2.26068.92805
Appendix 12 - Effective Acceleration and Gravitational Mediation in Reversible Mass-Energy Dynamics in ECM. DOI: https://doi.org/10.13140/RG.2.2.19018.48320
Appendix 13 - Proportionality Consistency and Inertial Balance. DOI: https://doi.org/10.13140/RG.2.2.25046.56648
Appendix 17 - Internal Force Generation and Non-Decelerative Dynamics in ECM under Negative Apparent Mass Displacement. DOI: https://doi.org/10.13140/RG.2.2.26584.61446
Appendix 23 - Frequency-Origin of Energy, Photon Kinetics, and Mass Displacement in ECM (Part-2: Frequency as more fundamental than mass). DOI: https://doi.org/10.13140/RG.2.2.17308.14723
Appendix 32 - Energy Density Structures in ECM. DOI: https://doi.org/10.13140/RG.2.2.22849.88168
Appendix 33 - Gravitating Mass and Its Polarity in Extended Classical Mechanics (ECM). DOI: https://doi.org/10.13140/RG.2.2.15395.16169
Appendix 34 - Scalar Mass Partitioning and Gravitational Phenomena beyond Relativity - the ECM Perspective. DOI: https://doi.org/10.13140/RG.2.2.32119.94881
Appendix 37 - Consistent Frequency-Energy-Radius Dynamics in ECM. DOI: http://dx.doi.org/10.13140/RG.2.2.21834.07362
Appendix 46 - Extended Classical Mechanics’ Elucidation - Newtonian Mechanics versus Relativistic Curvature in Explaining Dark Energy’s Dynamical Effects. DOI: https://doi.org/10.13140/RG.2.2.25127.41121
Appendix 47 - Variable Matter Mass in Extended Classical Mechanics (ECM). DOI: https://doi.org/10.13140/RG.2.2.20913.44643