The Toroidal Resonance Equation (TRE). A Theoretical Framework for Gravitational-Relativistic Oscillatory Dynamics

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Research Paper (postgraduate) from the year 2026 in the subject Physics, grade: A, , language: English, abstract: This paper introduces the Toroidal Resonance Equation (TRE), a mathematical model that extends classical gravitational acceleration with relativistic and oscillatory terms to describe radial dynamics in toroidal or resonant systems. The TRE incorporates a velocity-dependent relativistic factor, a sinusoidal time modulation, and a phenomenological field term raised to the power 2k. We derive its components, explore physical implications through analytical approximations and numerical methods, and discuss applications in astrophysics, plasma physics, and engineered resonance systems. Analytical solutions reveal oscillatory radial pulsations, while simulations demonstrate limit cycles and resonance amplification. Potential experimental validations using tokamak analogs and mechanical resonators are proposed to test predictions.

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