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Φ(t₀, τ) = |Σ_α w_α I_C(α, t₀, τ)| / [I_E(t₀, τ) · Σ_α w_α] I_E(t₀,τ) = [e^{−γt₀} − e^{−γ(t₀+τ)}] / (γτ) H = Σ p²/2m + ½Σ k_ij(q_i−q_j)² ⟨T⟩ = −½⟨Σ r_i · F_i⟩ Ω = Σ_α ω_α · w_α / Σ w_α

Key Equations

TRVIT Theorem 3.1:

$$\Phi(t_0,\tau) = \frac{|\sum_\alpha w_\alpha I_C(\alpha, t_0, \tau)|}{I_E(t_0, \tau) \cdot \sum_\alpha w_\alpha}$$

Energy Decay:

$$I_E(t_0,\tau) = \frac{e^{-\gamma t_0} - e^{-\gamma(t_0+\tau)}}{\gamma\tau}$$

Hamiltonian:

$$H = \sum \frac{p^2}{2m} + \frac{1}{2}\sum k_{ij}(q_i-q_j)^2$$

Virial (Clausius 1870):

$$\langle T \rangle = -\frac{1}{2}\langle \sum r_i \cdot F_i \rangle$$

GOMT Scalar:

$$\Omega = \frac{\sum_\alpha \omega_\alpha \cdot w_\alpha}{\sum w_\alpha}$$

Research Frameworks

Featured

TRVIT v2.2

Time-Resolved Virial Imbalance in N-Body Damped Harmonic Systems

April 2026 DOI: 10.5281/zenodo.19994556

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.

Main Result (Theorem 3.1):
$$\Phi(t_0,\tau) = \frac{|\sum_\alpha w_\alpha I_C(\alpha, t_0, \tau)|}{I_E(t_0, \tau) \cdot \sum_\alpha w_\alpha}$$
Zero-Overlap Index 1
Max Error 3.44×10⁻⁵
Gap Closed 156 years

Status: Preprint available on Zenodo (open access, CC BY 4.0). Journal submission status: contact for current details.

Read on Zenodo

JAP

The Jabbar Adal Principle

2026 DOI: 10.5281/zenodo.19815385

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.

Hamiltonian Mechanics Proved Theorems Phase-Space CC BY 4.0
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GOMT

Global Oscillator Mode Trace Framework

2026 DOI: 10.5281/zenodo.19798586

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.

Spectral Analysis Normal Modes Coupled Oscillators CC BY 4.0
Read on Zenodo