# Physics Equations Database — Mapped to Unified Equation

**Equations:** 333
**Equation:** Ω = Ψ [ B(θ) ⊗ C(n, α) ] ⊕ Δ(n, θ, α)

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## Eq 1. Newton's Three Laws of Motion

**Domain:** Classical Mechanics
**Description:** Foundation of all classical mechanics; inertial frames; F=dp/dt; action=reaction

| Symbol | Mapping |
|--------|---------|
| Ω | Force |
| Ψ | Laws of Motion |
| B | Mass and Inertia |
| C | External Forces and Acceleration |
| Δ | Friction and Air Resistance |

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## Eq 2. Lagrangian Mechanics (Principle of Least Action)

**Domain:** Classical Mechanics
**Description:** Action S=∫L dt; δS=0 → Euler-Lagrange equations

| Symbol | Mapping |
|--------|---------|
| Ω | Action S |
| Ψ | Lagrangian L |
| B | Conserved momentum p |
| C | Generalized coordinates q |
| Δ | Residual energy uncertainty |

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## Eq 3. Hamiltonian Mechanics

**Domain:** Classical Mechanics
**Description:** Canonical eqs: q̇=∂H/∂p, ṗ=−∂H/∂q; symplectic structure

| Symbol | Mapping |
|--------|---------|
| Ω | Hamiltonian |
| Ψ | Lagrangian/Hamiltonian operator |
| B | Symplectic basis |
| C | External potential/force |
| Δ | Thermal noise/residual error |

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## Eq 4. Hamilton-Jacobi Equation

**Domain:** Classical Mechanics
**Description:** ∂S/∂t + H(q,∂S/∂q,t)=0; bridges classical→quantum

| Symbol | Mapping |
|--------|---------|
| Ω | Action S |
| Ψ | Hamiltonian H |
| B | Phase space coordinates q |
| C | Time t and generalized momenta ∂S/∂q |
| Δ | Residual energy uncertainty |

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## Eq 5. Euler-Lagrange Equation

**Domain:** Classical Mechanics
**Description:** d/dt(∂L/∂q̇) − ∂L/∂q = 0; from δS=0

| Symbol | Mapping |
|--------|---------|
| Ω | Lagrangian |
| Ψ | Hamiltonian |
| B | Kinetic energy |
| C | Potential energy |
| Δ | Friction |

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## Eq 6. D'Alembert's Principle

**Domain:** Classical Mechanics
**Description:** Virtual work for dynamics: Σ(F_i−ṗ_i)·δr_i=0

| Symbol | Mapping |
|--------|---------|
| Ω | Virtual work |
| Ψ | D'Alembert's operator |
| B | Conserved forces |
| C | External dynamics |
| Δ | Residual forces |

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## Eq 7. Euler's Rigid Body Rotation Equations

**Domain:** Classical Mechanics
**Description:** I·ω̇ + ω×(I·ω) = τ; angular momentum dynamics

| Symbol | Mapping |
|--------|---------|
| Ω | Angular momentum |
| Ψ | Euler's rotation equations |
| B | Inertia tensor (I) |
| C | External torque (τ) |
| Δ | Frictional losses |

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## Eq 8. Conservation of Momentum

**Domain:** Classical Mechanics
**Description:** dP/dt = ΣF_ext; P constant when ΣF_ext=0

| Symbol | Mapping |
|--------|---------|
| Ω | Momentum |
| Ψ | Newton's second law |
| B | Mass |
| C | External forces |
| Δ | Frictional losses |

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## Eq 9. Conservation of Angular Momentum

**Domain:** Classical Mechanics
**Description:** dL/dt = τ_ext; L=Iω constant when τ=0

| Symbol | Mapping |
|--------|---------|
| Ω | Angular Momentum |
| Ψ | Mechanical System |
| B | Moment of Inertia |
| C | External Torque |
| Δ | Frictional Loss |

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## Eq 10. Conservation of Energy

**Domain:** Classical Mechanics
**Description:** dE/dt=0 for isolated system; time translation symmetry

| Symbol | Mapping |
|--------|---------|
| Ω | Total Energy |
| Ψ | Lagrangian or Hamiltonian |
| B | Kinetic and Potential Energies |
| C | External Forces and Constraints |
| Δ | Thermal Fluctuations |

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## Eq 11. Work-Energy Theorem

**Domain:** Classical Mechanics
**Description:** W=ΔKE; ∫F·dr = ½mv²_f − ½mv²_i

| Symbol | Mapping |
|--------|---------|
| Ω | Kinetic Energy |
| Ψ | Classical Mechanics Theory |
| B | Mass (m) |
| C | Force (F) and Displacement (r) |
| Δ | Initial Kinetic Energy |

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## Eq 12. Impulse-Momentum Theorem

**Domain:** Classical Mechanics
**Description:** J=∫F dt=Δp

| Symbol | Mapping |
|--------|---------|
| Ω | Change in momentum |
| Ψ | Force applied over time |
| B | Conserved momentum basis |
| C | External force or torque |
| Δ | Uncertainty or residual error |

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## Eq 13. Center of Mass Equation

**Domain:** Classical Mechanics
**Description:** MR̈_cm=ΣF_ext; COM moves like point particle

| Symbol | Mapping |
|--------|---------|
| Ω | Position of Center of Mass |
| Ψ | Newton's Second Law |
| B | Mass of System |
| C | External Forces Acting on System |
| Δ | Uncertainty in Position |

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## Eq 14. Hooke's Law

**Domain:** Continuum Mechanics
**Description:** F=−kx; σ=Eε; linear elastic response

| Symbol | Mapping |
|--------|---------|
| Ω | Force |
| Ψ | Hooke's Law Operator |
| B | Spring Constant |
| C | Displacement |
| Δ | Viscoelastic Loss |

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## Eq 15. Parallel Axis Theorem

**Domain:** Classical Mechanics
**Description:** I=I_cm+Md²

| Symbol | Mapping |
|--------|---------|
| Ω | Moment of inertia |
| Ψ | Parallel Axis Theorem |
| B | Moment of inertia at center of mass |
| C | Distance from center of mass squared |
| Δ | Residual moment of inertia |

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## Eq 16. Coriolis Force

**Domain:** Classical Mechanics
**Description:** F_cor=−2m ω×v' (rotating frame)

| Symbol | Mapping |
|--------|---------|
| Ω | Coriolis Force |
| Ψ | Classical Mechanics Operator |
| B | Angular Velocity (ω) |
| C | Velocity in Rotating Frame (v') |
| Δ | Measurement Uncertainty |

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## Eq 17. Centrifugal Force

**Domain:** Classical Mechanics
**Description:** F_cf=−m ω×(ω×r) (rotating frame)

| Symbol | Mapping |
|--------|---------|
| Ω | Centrifugal Force |
| Ψ | Classical Mechanics Theory |
| B | Angular Velocity Vector (ω) |
| C | Radial Distance from Axis (r) |
| Δ | Measurement Uncertainty |

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## Eq 18. Simple Harmonic Motion

**Domain:** Classical Mechanics
**Description:** ẍ+ω²x=0; x=A cos(ωt+φ); T=2π/ω

| Symbol | Mapping |
|--------|---------|
| Ω | Displacement |
| Ψ | Simple Harmonic Motion Theory |
| B | Conserved Angular Frequency |
| C | External Force or Damping |
| Δ | Energy Loss or Friction |

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## Eq 19. Damped Harmonic Oscillator

**Domain:** Classical Mechanics
**Description:** ẍ+2βẋ+ω₀²x=0; under/over/critically damped

| Symbol | Mapping |
|--------|---------|
| Ω | Position or displacement of the oscillator |
| Ψ | Differential equation describing the system's dynamics |
| B | Spring constant, fundamental property of the oscillator |
| C | Friction coefficient, external damping force |
| Δ | Energy loss due to friction and other dissipative forces |

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## Eq 20. Forced Oscillator + Resonance

**Domain:** Classical Mechanics
**Description:** ẍ+2βẋ+ω₀²x=(F₀/m)cos ωt; A=F₀/m/√((ω₀²−ω²)²+4β²ω²)

| Symbol | Mapping |
|--------|---------|
| Ω | Displacement of the oscillator |
| Ψ | Forced Oscillator + Resonance theory |
| B | Conserved basis: mass and spring constant |
| C | Dynamic context: external force amplitude and frequency |
| Δ | Residual error: damping coefficient uncertainty |

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