TRVIT Theorem 3.1:
Energy Decay:
Hamiltonian:
Virial (Clausius 1870):
GOMT Scalar:
Independent Theoretical Physicist
Closed-form analytical results in classical mechanics, virial theory, and Hamiltonian mechanics
Muhammad Umar Jabbar is an independent theoretical physicist working from Khanewal, Punjab, Pakistan. Without institutional affiliation or external funding, he has produced closed-form analytical results in classical mechanics that had not appeared anywhere in the literature since Clausius established the Virial Theorem in 1870.
His flagship result — Theorem 3.1 of TRVIT v2.2 — is a closed-form expression for the normalised kinetic-potential imbalance Φ(t₀, τ) in a deterministic N-body coupled damped harmonic system over an arbitrary finite observation window, extending the classical virial theorem to this driven-damped setting. The result is numerically verified to absolute error ≤ 3.44 × 10⁻⁵ against an O(ε²) analytical error bound.
This work was achieved through pure mathematical reasoning, from home, in one of Pakistan's smaller cities — demonstrating that the highest level of theoretical physics is determined by mathematical truth, not institutional prestige or national geography.
"A result is useful when it is exact, or when its error is bounded. The Zero-Overlap Index is not self-promotion — it is a falsifiable claim that shifts the burden of proof appropriately."
Phase-space formulations, normal-mode decomposition, Liouville volume contraction, and canonical transformations for non-relativistic systems.
Finite-window imbalance, kinetic-potential equipartition in dissipative systems, and exact analytical window-length criteria since Clausius 1870.
Proportional, Rayleigh, and weakly non-proportional damping. Formal O(ε²) error bounds for weak-damping approximations in N-body chains.
Coupled chains, mass-stiffness eigenproblems, mode-weight interference, spectral fingerprinting via the GOMT single-scalar framework.
Author list removed for double-blind review · Submitted to Physica Scripta
The classical virial theorem holds only for conservative power-law potentials; it breaks down once damping and periodic forcing enter, as in almost every real driven oscillator. This paper derives an exact identity for the time-averaged energy partition of a periodically driven, linearly damped, non-harmonic oscillator in its non-equilibrium steady state (NESS), adding a damping cross-correlation term and a forcing cross-correlation term to the classical result. The identity is verified by adaptive eighth-order Runge–Kutta integration (SciPy DOP853) across thirty independent parameter combinations, with residuals below 0.13% — consistent with floating-point precision. The two correction terms define a measurable scalar, the virial defect Δ, computable directly from a single time series without knowledge of the potential's coefficients.
Time-Resolved Virial Imbalance in N-Body Damped Harmonic Systems
Exact Closed-Form Expression, Formal Error Bounds, Symmetry Analysis, and Zero-Overlap Index. The first closed-form expression for the normalised kinetic-potential imbalance in N-body coupled damped harmonic systems.
Status: Preprint available on Zenodo (open access, CC BY 4.0). Journal submission status: contact for current details.
The Jabbar Adal Principle
Two formally proved theorems in non-relativistic Hamiltonian mechanics, establishing new structural symmetry constraints on phase-space trajectories. Exact necessary conditions for canonical transformations that preserve action-angle variable structure.
Read on ZenodoGlobal Oscillator Mode Trace Framework
A single-scalar spectral fingerprint for N coupled harmonic oscillators. Collapses the full normal-mode spectrum into one invariant scalar, enabling mode identification, degeneracy detection, and structural comparison with O(1) storage.
Read on ZenodoExact identity for the time-averaged energy partition of a periodically driven, damped, non-harmonic oscillator in its non-equilibrium steady state. Introduces the virial defect Δ, verified to within 0.13% across 30 parameter combinations.
Exact closed-form expression for normalised kinetic-potential imbalance in N-body coupled damped harmonic systems. Error bound ≤ 3.44 × 10⁻⁵, O(ε²) proved.
Two formally proved theorems in non-relativistic Hamiltonian mechanics establishing structural symmetry constraints on phase-space trajectories.
Single-scalar spectral fingerprint for N coupled harmonic oscillators. Enables mode identification and structural comparison with O(1) storage.
Finite-window imbalance left unsolved for 156 years.
Global Oscillator Mode Trace framework on Zenodo. DOI: 10.5281/zenodo.19798586
Jabbar Adal Principle on Zenodo. DOI: 10.5281/zenodo.19815385
Submitted to Journal of Physics A (IOP Publishing). Zenodo DOI: 10.5281/zenodo.19994556
Preprint deposited on Zenodo, open access, CC BY 4.0.
Clausius published the Virial Theorem in 1870. For 156 years, no one derived the finite-window virial imbalance in closed form for N-body damped systems. TRVIT Theorem 3.1 is the first such result.
Means the result does not overlap with any prior literature, provable by citation analysis. All three frameworks carry ZOI = 1.
All three frameworks developed independently, without institutional affiliation, without grants, and without co-authors. Pure mathematical reasoning from home.
All publications open access on Zenodo under CC BY 4.0. No paywalls. Reproducibility and accessibility at the core.
TRVIT v2.2 deposited as an open-access preprint on Zenodo. Formal journal submission status available on request.
Every theorem stated with explicit hypotheses, formal proofs, and numerical verification. Error bounds formally proved and empirically validated.
Independent theoretical physicist from Khanewal, Punjab, Pakistan, working outside any university affiliation. Author of the TRVIT, JAP, and GOMT frameworks in classical mechanics, published open access on Zenodo in 2026. No institutional affiliation, no external funding, no co-authors.
A closed-form expression for the normalised kinetic-potential imbalance Φ(t₀, τ) in an N-body coupled damped harmonic oscillator system over any finite observation window, extending virial-theorem analysis (originating with Clausius, 1870) to this driven-damped regime. Error bound ≤ 3.44 × 10⁻⁵ against a formally derived O(ε²) bound.
A falsifiability metric for scientific novelty. ZOI of 1 means no prior literature contains the same result. All three frameworks carry ZOI = 1, provable through systematic citation analysis and literature review.
All freely available on Zenodo under CC BY 4.0. TRVIT: doi.org/10.5281/zenodo.19994556 · JAP: doi.org/10.5281/zenodo.19815385 · GOMT: doi.org/10.5281/zenodo.19798586
Two formally proved theorems in non-relativistic Hamiltonian mechanics establishing structural symmetry constraints on phase-space trajectories. Provides exact necessary conditions for canonical transformations that preserve the action-angle variable structure, proved from first principles without approximation.
Global Oscillator Mode Trace — a single-scalar spectral fingerprint for N coupled harmonic oscillators. Collapses the full normal-mode spectrum into one invariant scalar, enabling mode identification, degeneracy detection, and structural comparison with O(1) storage. Analytically derived with exact closed-form expressions.
Every publication includes explicit hypotheses, formal proofs, and numerical verification code. Error bounds are formally proved and empirically validated. All code and data are reproducible from the Zenodo repositories. The Zero-Overlap Index claim is falsifiable through systematic literature review.
Matriculation (Science Group) from Govt. High School 2/AH, Khanewal, BISE Multan, 2022 — 1006/1100. Intermediate Part-II (Pre-Medical) from Superior College for Boys, Khanewal, BISE Multan, 2024 — 858/1200. Full subject-wise breakdown is on the Education page.
No. All research here is produced independently, without university affiliation, grant funding, or co-authors. This is stated explicitly in every manuscript's acknowledgments and conflict-of-interest sections.
Status varies by piece and is stated on each page. The virial identity manuscript is currently submitted to Physica Scripta and under double-blind review — it has not yet been accepted or formally published. Zenodo deposits are open-access preprints with a DOI but are not peer-reviewed by Zenodo itself. Peer-review status is never implied where it hasn't happened.
Python with SciPy's solve_ivp, typically the adaptive eighth-order Runge–Kutta method DOP853, at tight absolute/relative tolerances (10⁻¹² / 10⁻¹⁰). Verification runs are independently re-implemented from scratch as a second check before results are reported.
Yes, in places — for drafting assistance, code scaffolding, and website development, similar to how many independent researchers use modern tooling. Any manuscript that involved substantial AI drafting assistance discloses that explicitly in its authorship or acknowledgments rather than presenting it as unaided work. The mathematics and numerical results themselves are independently checkable regardless of what tools helped produce the prose around them.
Yes. Zenodo-deposited work has a citable DOI (see the Publications page). The Physica Scripta submission should be cited as "manuscript submitted for review" until a formal citation is available, in line with normal academic convention for unpublished work under review.
Independent, open-access publication (Zenodo, arXiv-style preprints) lowers the barrier for researchers without institutional access while still allowing formal journal submission in parallel. It does not replace peer review — it makes work available for scrutiny while that process runs.
English for technical writing and research; Urdu for translation work and some creative writing. Bilingual content is noted where it appears on the site.
Email umarjaumofficial@gmail.com, or use the form on the Contact page. Review requests, error reports, and collaboration proposals are all welcome — corrections to published work are taken seriously and will be acknowledged.
Primarily classical and Hamiltonian mechanics — virial-theorem identities, N-body oscillator systems, and related closed-form derivations — with numerical verification via adaptive Runge–Kutta integration. See the Research page for the full portfolio.
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Submitted to Physica Scripta and currently under double-blind review, as of 2026. This page will be updated when a decision is issued — no earlier claim of acceptance is made or implied.
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