The Jabbar Adal Principle

The Jabbar Adal Principle (JAP) is a theoretical framework in non-relativistic Hamiltonian mechanics proposed by Muhammad Umar Jabbar, an independent researcher from Khanewal, Punjab, Pakistan, in April 2026. The principle formally names, defines, and analyses the instantaneous ratio of kinetic energy to potential energy at every local subsystem of a closed physical system — a quantity that existing laws of thermodynamics and the virial theorem do not address. It introduces the Jabbar Asymmetry Parameter A(i, t), the Adal Asymmetry Index AAI(t), and proposes a new named dimensionless unit, the Jabbar (symbol: Jb), named after the author's family — derived from his father, Abdul Jabbar, of village 4/A.H, Khanewal, Punjab, Pakistan.

The JAP comprises two proved theorems: the Jabbar Vitality Theorem, which gives a necessary and sufficient scalar condition for physical processes to be possible in a closed system; and Theorem R1 (the Oscillatory Modulation Theorem), which derives that the Adal Asymmetry Index of a coupled oscillator system oscillates at a beat frequency determined entirely by the Hamiltonian eigenvalues. The work was publicly deposited on Zenodo (CERN-hosted) with DOI 10.5281/zenodo.19815385 and ORCID 0009-0008-5968-0991, and is licensed under CC BY 4.0. It is currently pending formal peer review.

Background

Unanswered question in energy law

The First law of thermodynamics states that the total energy of a closed system is conserved: dE/dt = 0. The Second law of thermodynamics states that entropy does not decrease: dS/dt ≥ 0. The virial theorem, due to Rudolf Clausius (1870), gives the time-averaged ratio ⟨T⟩ = (n/2)⟨V⟩ for a stationary system under a potential proportional to xⁿ.

None of these results names or analyses the instantaneous ratio T_i(t)/|V_i(t)| at every local subsystem, nor its deviation from the virial stationary value, as a computable diagnostic quantity. The JAP was proposed to formally address this gap within the domain of closed, non-relativistic Hamiltonian systems.

Domain of validity

The JAP applies exclusively to closed physical systems governed by a non-relativistic Hamiltonian H = T + V, where T ≥ 0 is kinetic energy and V is potential energy referenced to zero at infinite separation. It explicitly excludes general relativity (where gravitational energy cannot be localised), quantum field theory (infinite vacuum energy before renormalisation), and cosmological scales. The domain encompasses molecular vibrations, coupled mechanical resonators, acoustic lattices, optical traps, and any classical N-body system in the Newtonian limit.

The Jabbar unit [Jb]

The Jabbar is a named dimensionless unit, analogous to the radian (named dimensionless angle) and the neper (named dimensionless logarithmic ratio). It is named after the Jabbar family of Khanewal, Punjab, Pakistan — specifically derived from Abdul Jabbar, father of the author and a daily-wage labourer, whose sacrifices made the research possible. The unit was introduced in April 2026 and registered in the scientific record via Zenodo/CERN.

JAP quantities and their units

Quantity	Symbol	Unit	Range
Jabbar Asymmetry Parameter	A(i, t)	Jb	[0, ∞)
Virial Deviation Parameter	ΔA(i, t)	Jb	(−∞, +∞)
Adal Asymmetry Index	AAI(t)	Jb	[0, ∞)
Virial stationary value	Avir(i)	1.000 Jb	system-specific

At thermodynamic equilibrium: AAI = 0 Jb. For a maximally non-equilibrium system: AAI → ∞ Jb.

Formal definitions

The system considered is N coupled harmonic oscillators governed by the Hamiltonian:

$H=T+V=\sum _{i=1}^N\frac{p_i^2}{2m_i}+\frac{1}{2}{k}_i^{\mathrm{e}\mathrm{f}\mathrm{f}}{x}_i^2-\kappa x_1{x}_2[\mathrm{J}]$

Jabbar Asymmetry Parameter

For subsystem i at time t, with V_i(t) ≠ 0:

$A(i,t)=\frac{T_i(t)}{|V_i(t)|}[\mathrm{J}\mathrm{b}]$

Virial Deviation Parameter

$\Delta A(i,t)=A(i,t)-A_{\mathrm{v}\mathrm{i}\mathrm{r}}(i)[\mathrm{J}\mathrm{b}]$

where for a quadratic potential (n = 2), the virial theorem gives A_vir = 1.000 Jb.

Adal Asymmetry Index

$\mathrm{A}\mathrm{A}\mathrm{I}(t)=\frac{1}{N}\sum _{i=1}^N|\Delta A(i,t)|[\mathrm{J}\mathrm{b}]$

AAI = 0 Jb if and only if every subsystem satisfies the virial theorem exactly and simultaneously (thermodynamic equilibrium).

The two theorems

Jabbar Vitality Theorem

Theorem 1 (Jabbar Vitality Theorem). In any closed non-relativistic Hamiltonian system S, physical processes are possible if and only if AAI(t) > 0 Jb.

The proof proceeds by biconditional. (⇒) If AAI(t) > 0, there exists a subsystem i not at its virial stationary state, so energy gradients exist and processes occur. (⇐) If processes occur, energy gradients exist, so some subsystem is not at virial equilibrium, so AAI(t) > 0.

The theorem is logically independent of the Second Law: the Second Law gives the direction of spontaneous processes (dS/dt ≥ 0); the Jabbar Vitality Theorem gives the necessary and sufficient scalar condition for processes to be possible at all, in terms of a directly computable quantity AAI [Jb].

Theorem R1 — Oscillatory Modulation Theorem

ω₂ − ω₁

— is determined by the eigenvalues of the stiffness matrix K.

The derivation proceeds from Hamilton's equations (1835) via normal-mode decomposition and the product-to-sum identity:

$\sin ({\omega }_{1}t+{\varphi }_1)\cdot \sin ({\omega }_{2}t+{\varphi }_2)=\frac{1}{2}[\cos ({\omega }_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{t}}t+\psi )-\cos (({\omega }_1+{\omega }_2)t+{\psi }^{\prime })]$

The cross-mode beat term at ω_beat = |ω₂ − ω₁| is the mathematical origin of the oscillatory modulation in AAI(t). The heat equation predicts no such modulation (AAI_heat(t) = AAI₀·e^−γt), as it does not track T_i/|V_i| within each subsystem.

Numerical verification

The two-oscillator system used for verification has parameters: m₁ = 1.0 kg, m₂ = 2.0 kg, k₁ = 4.0 N m⁻¹, k₂ = 6.0 N m⁻¹, coupling stiffness κ = 1.5 N m⁻¹, and damping coefficient γ = 0.02 s⁻¹.

Hamilton's equations were integrated using the Dormand–Prince DOP853 method (rtol = 10⁻¹¹, atol = 10⁻¹⁴). Independently verified by Python/NumPy and a supplementary PHP 8.3 implementation (JAP_calculations.php, CC BY 4.0).

Key verified numerical results

Quantity	Theoretical	Numerical (fitted)	Deviation
ω1	1.802776 rad s−1	1.802776 rad s−1	0%
ω2	2.449490 rad s−1	2.449490 rad s−1	0%
ωbeat	0.646714 rad s−1	0.646686 rad s−1	0.0043%
Tbeat	9.7156 s	—	—
AAI(0)	1.000 Jb	1.000 Jb	0%
RMSE improvement over heat equation	37.54%

The beat frequency ω_beat is recovered from the AAI(t) signal to 1 part in 23,000 — consistent with an eigenvalue of the physical system, not a free fit parameter.

Comparison with established laws

Comparison of JAP with established energy laws

Law	What it states	What it does not address	JAP contribution
First Law (1865)	dE/dt = 0	Vi| ratio	Quantities A, AAI
Second Law (1865)	dS/dt ≥ 0	Biconditional for activity	Vitality Theorem
Virial theorem (1870)	⟨T⟩ = (n/2)⟨V⟩	Instantaneous deviations; named scalar	ΔA, Jb unit
Gibbs free energy (1876)	G = H − TS at const. T, P	All conditions; ratio diagnostics	Domain-universal
JAP (2026)	ΔA ≠ 0 for active systems; R1 modulation	—	Named quantities & Jb unit

Open problem: JAP-Entropy Conjecture

The author states an unproved conjecture: in the zero-temperature limit (T_bath → 0), during monotonic relaxation:

$\mathrm{A}\mathrm{A}\mathrm{I}(t)\propto \frac{{\sigma }_{\mathrm{e}\mathrm{p}}(t)}{⟨T(t)⟩}$

where σ_ep is the entropy production rate. If proved, this would connect AAI directly to irreversibility and non-equilibrium thermodynamics. The conjecture is the highest-value open problem identified within the JAP framework.

Priority and registration

Priority is established by public deposit on Zenodo (CERN-hosted) with DOI 10.5281/zenodo.19815385, ORCID 0009-0008-5968-0991, dated April 2026. Under CC BY 4.0, any use of the named quantities (A, ΔA, AAI, Jb) requires citation of the original work.

The following elements are claimed as novel: (1) the quantity A(i,t) = T_i/|V_i| as a named observable; (2) the Virial Deviation Parameter ΔA(i,t); (3) the Adal Asymmetry Index AAI(t); (4) the Jabbar unit [Jb]; (5) the Jabbar Vitality Theorem; (6) Theorem R1.

See also

• Virial theorem
• Hamiltonian mechanics
• Non-equilibrium thermodynamics
• First law of thermodynamics
• Second law of thermodynamics
• Muhammad Umar Jabbar

External links

• Full paper on Zenodo (DOI: 10.5281/zenodo.19815385)
• Author ORCID profile

Category:Hamiltonian mechanics Category:Non-equilibrium thermodynamics Category:Classical mechanics Category:Physics theorems Category:2026 in science

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