Research-Stack/3-Mathematical-Models/physics_eqs_mapped.md

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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