Yukawa Phase-Space Frontier

Explore the evolution of physical systems from classical orbital precession and spiral collapse to quantum phase shifts, global state-space mapping, and energy-preserving geometric integration.

https://youtu.be/GmEkJKmF1nk

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🍄Deliverables 21

Precession and Orbital Collapse in Screened Potentials

https://payhip.com/b/BacMg

Documents

Computational and numerical models
Snippets
An instructional overview and conceptual guide to screened physics, explaining how environmental screening factors alter standard gravitational and electromagnetic mechanics.

Exploration Frontiers

Precessing Bound Orbits (Rosette Dynamics) $\rightarrow$ Perturbation Theory for Periastron Advance
Unstable Dynamics & Spiral Collapse $\rightarrow$ Bifurcation Analysis of the Effective Potential
High-Energy Repulsive Scattering $\rightarrow$ Quantum Mechanical S-Wave Scattering
Dissipative Energy Loss / Drag $\rightarrow$ Phase Space Velocity & Energy-Momentum Diagrams
Standard Euler/RK ODE Integrators $\rightarrow$ Symplectic Integration Schemes (Velocity Verlet / Yoshida)

graph TD
    %% Class Definitions for Deep Saturated Dark Mode Palette
    classDef currentStyle fill:#111827,stroke:#3B82F6,stroke-width:1.5px,color:#93C5FD;
    classDef currentNode fill:#1F2937,stroke:#60A5FA,stroke-width:1px,color:#F9FAFB;
    
    classDef foundationStyle fill:#2E1065,stroke:#F59E0B,stroke-width:2px,color:#FDE68A;
    classDef outerFutureStyle fill:#0F172A,stroke:#0284C7,stroke-width:2px,color:#38BDF8;
    
    classDef analyticalStyle fill:#064E3B,stroke:#10B981,stroke-width:1.5px,color:#6EE7B7;
    classDef analyticalNode fill:#022C22,stroke:#34D399,stroke-width:1px,color:#F0FDF4;
    
    classDef modelsStyle fill:#0C4A6E,stroke:#0284C7,stroke-width:1.5px,color:#7DD3FC;
    classDef modelsNode fill:#082F49,stroke:#38BDF8,stroke-width:1px,color:#F0F9FF;
    
    classDef visualsStyle fill:#4C0519,stroke:#E11D48,stroke-width:1.5px,color:#FDA4AF;
    classDef visualsNode fill:#2E0616,stroke:#FB7185,stroke-width:1px,color:#FFF1F2;

    %% Base: Current Exploration Regimes
    subgraph Current_Exploration [CURRENT EXPLORATION REGIMES]
        style Current_Exploration fill:#111827,stroke:#3B82F6,stroke-width:1.5px,color:#93C5FD;
        C1["Bound Orbits:<br/>Rosette Dynamics"]:::currentNode
        C2["Unstable Dynamics:<br/>Spiral Collapse"]:::currentNode
        C3["High-Energy Dynamics:<br/>Repulsive Scattering"]:::currentNode
    end

    %% Transition: Foundational Physics
    T1["<b>ESTABLISHED MODELS & NUMERICS</b><br/>Yukawa Potential: <i>V(r) = -A exp(-αr) / r</i><br/>Standard ODE Integrators"]:::foundationStyle

    Current_Exploration -.-> T1

    %% Target: Potential Future Exploration Roadmap
    subgraph Potential_Orientation [POTENTIAL FUTURE EXPLORATION ROADMAP]
        style Potential_Orientation fill:#0F172A,stroke:#0284C7,stroke-width:2px,color:#38BDF8;
        direction TB

        subgraph Analytical [1. Analytical Phase Space & Stability]
            style Analytical fill:#064E3B,stroke:#10B981,stroke-width:1.5px,color:#6EE7B7;
            A1["Poincaré Maps:<br/>Quasi-Periodic vs Chaotic Transition"]:::analyticalNode
            A2["Bifurcation of Veff Peak:<br/>Identify Critical Parameters αc, Lc"]:::analyticalNode
            A3["Perturbation Theory:<br/>Derive Analytical Periastron Advance Δφ"]:::analyticalNode
        end

        subgraph Models [2. Enhanced Numerical & Physical Models]
            style Models fill:#0C4A6E,stroke:#0284C7,stroke-width:1.5px,color:#7DD3FC;
            M1["Symplectic Integration:<br/>Velocity Verlet for Long-Term Stability"]:::modelsNode
            M2["3D Motion:<br/>Out-of-Plane Perturbations / Anisotropy"]:::modelsNode
            M3["Quantum Scattering:<br/>S-Wave Scattering & Phase Shift Extraction"]:::modelsNode
        end

        subgraph Visuals [3. Extended Diagnostic Visualizations]
            style Visuals fill:#4C0519,stroke:#E11D48,stroke-width:1.5px,color:#FDA4AF;
            V1["Phase Space Velocity Maps:<br/>Overlaid (r, r') Fields"]:::visualsNode
            V2["(E, L) Phase Diagram:<br/>Map Outcome Zones (Bound, Collapse, Unbounded)"]:::visualsNode
        end
    end

    T1 ==> Analytical
    T1 ==> Models
    T1 ==> Visuals

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