Editing Openai/674467e2-4648-800d-8ea7-f9a78979c12c (section)

===== Key Concepts in Mathematical Form =====
# Neutral Centers as Focal Points FNC=∑i=1nϕi⋅ψiF_{NC} = \sum_{i=1}^n \phi_i \cdot \psi_iFNC=i=1∑nϕi⋅ψi - FNCF_{NC}FNC: Force or energy converging at the neutral center. - ϕi,ψi\phi_i, \psi_iϕi,ψi: Vibrational potentials from interacting masses iii, in harmonic resonance.
# Levitation via Cycloid Motion Cycloid motion is modeled as: x(t)=r(t−sin⁡(t)),y(t)=r(1−cos⁡(t))x(t) = r(t - \sin(t)), \quad y(t) = r(1 - \cos(t))x(t)=r(t−sin(t)),y(t)=r(1−cos(t)) - x(t),y(t)x(t), y(t)x(t),y(t): Position of a point on the cycloid path. - rrr: Radius of the generating circle. - This motion minimizes resistance and enhances energy transfer, aligning with Schauberger's levitative forces.
# Gravity and Gravitism Keely’s gravity as a sympathetic stream: G=∫V∇⋅E dVG = \int_{V} \nabla \cdot \mathbf{E} \, dVG=∫V∇⋅EdV - GGG: Gravitational force as the flux divergence of the etheric field E\mathbf{E}E through a volume VVV. Russell’s view of gravity as centripetal compression: G(r)=kr2⋅exp⁡(−αr)G(r) = \frac{k}{r^2} \cdot \exp(-\alpha r)G(r)=r2k⋅exp(−αr) - G(r)G(r)G(r): Gravitational potential, modified by the exponential decay exp⁡(−αr)\exp(-\alpha r)exp(−αr) to account for vibrational attenuation. - kkk: Gravitational constant. - α\alphaα: Vibratory damping factor.
# Levitation and Magnetolytic Dissociation Schauberger’s levitation through magnetism: Flev=q⋅(v×B)F_{lev} = q \cdot (\mathbf{v} \times \mathbf{B})Flev=q⋅(v×B) - FlevF_{lev}Flev: Levitational force. - qqq: Charge of ions involved in levitative processes. - v\mathbf{v}v: Velocity of the ions. - B\mathbf{B}B: Magnetic field intensity. Magnetolytic dissociation intensifies levitation: ΔFlev=Flev⋅(1+Δvv)\Delta F_{lev} = F_{lev} \cdot \left(1 + \frac{\Delta v}{v}\right)ΔFlev=Flev⋅(1+vΔv) - ΔFlev\Delta F_{lev}ΔFlev: Enhanced levitation force. - Δv\Delta vΔv: Incremental velocity from rotational acceleration.
# Energy Liberation in Molecular Aggregation Keely’s latent energy liberation: Elatent=∑i=1nki⋅ωi2E_{latent} = \sum_{i=1}^n k_i \cdot \omega_i^2Elatent=i=1∑nki⋅ωi2 - ElatentE_{latent}Elatent: Latent energy stored in molecular aggregation. - kik_iki: Spring constant or vibratory stiffness of the iii-th bond. - ωi\omega_iωi: Angular velocity of rotational molecular vibrations.
# Sympathetic Vibratory Forces Sympathetic vibration as a resonance phenomenon: ω=km\omega = \sqrt{\frac{k}{m}}ω=mk - ω\omegaω: Natural resonant frequency. - kkk: Restoring vibratory force constant. - mmm: Effective mass. Energy transfer through sympathetic induction: Eind=A⋅sin⁡(ωt+ϕ)E_{ind} = A \cdot \sin(\omega t + \phi)Eind=A⋅sin(ωt+ϕ) - EindE_{ind}Eind: Induced energy. - AAA: Amplitude of excitation. - ϕ\phiϕ: Phase offset for harmonic alignment.
# Implosion vs. Explosion Dynamics Schauberger’s implosion dynamics: Pimpl=12ρv2−kTP_{impl} = \frac{1}{2} \rho v^2 - \frac{k}{T}Pimpl=21ρv2−Tk - PimplP_{impl}Pimpl: Implosive pressure. - ρ\rhoρ: Density of the medium. - vvv: Velocity of inward flow. - TTT: Temperature. - kkk: Proportional constant. Explosion dynamics (destructive forces): Pexpl=ρv2+kTP_{expl} = \rho v^2 + \frac{k}{T}Pexpl=ρv2+Tk
# Gravitational and Levitative Forces as Dualities Expressing Russell’s dualities: Fgrav+Flev=0F_{grav} + F_{lev} = 0Fgrav+Flev=0 Where: Fgrav=−∇⋅Egrav,Flev=∇⋅ElevF_{grav} = -\nabla \cdot \mathbf{E}_{grav}, \quad F_{lev} = \nabla \cdot \mathbf{E}_{lev}Fgrav=−∇⋅Egrav,Flev=∇⋅Elev - FgravF_{grav}Fgrav: Gravitational force as centripetal. - FlevF_{lev}Flev: Levitational force as centrifugal.
# Efficiency of Implosion-Based Machines Schauberger’s efficiency formula: η=1−RresistRform\eta = 1 - \frac{R_{resist}}{R_{form}}η=1−RformRresist - η\etaη: Efficiency. - RresistR_{resist}Rresist: Resistance to natural motion. - RformR_{form}Rform: Formative energy contribution.
# Vibrational Energy Density Keely’s vibrational energy density: U=12k⋅x2U = \frac{1}{2} k \cdot x^2U=21k⋅x2 - UUU: Energy density. - xxx: Displacement of the system in vibratory motion.