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Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>
99 lines
3.3 KiB
Markdown
99 lines
3.3 KiB
Markdown
# Universal Equations
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> Page 23. Three Tests. Euler-Lagrange. The backbone.
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>
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> 📖 **Key research:** Euler (1744) and Lagrange (1755) independently derived the variational equations. The principle of stationary action — that nature takes the path of least action — underlies *all* of classical mechanics, electromagnetism, relativity, and quantum field theory. One equation to rule them all. Noether (1915) then showed that every symmetry of the action corresponds to a conservation law ([Noether's theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem)).
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## The Three Tests (§177)
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A universal equation must pass all three:
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| Test | Criterion | QWERTY |
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|------|-----------|--------|
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| 1 | It governs many systems → **SCOPE** | SCOPE = 56 |
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| 2 | It falls out of symmetry or variational principles → **STRUCTURE** | STRUCTURE = SHELL = 69 |
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| 3 | It reduces to known special cases without breaking → **LIMITS** | LIMITS = TRIVIAL = 78 |
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```
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TESTS = REAL = 37 prime (what passes the tests = real)
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GOVERN = CREATIVE = MARCH = 79 prime (to govern = creative)
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LIMITS = TRIVIAL = BINARY = 78 (the limits are trivial)
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SYMMETRY = OPTIMAL = CRITERION = 88 (Noether: symmetry = conservation)
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```
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**The Euler-Lagrange equation passes all three.**
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---
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## Principle of Stationary Action (§177)
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```
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δS = 0, S = ∫ L(q,q̇,t) dt
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```
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The physical path is where the action S does not change to first order.
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**Euler-Lagrange equations:**
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```
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d/dt(∂L/∂q̇ᵢ) − ∂L/∂qᵢ = 0
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```
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**Field form:**
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```
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∂_μ(∂L/∂(∂_μφₐ)) − ∂L/∂φₐ = 0
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```
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*Her note: "This is the backbone. Choose the right Lagrangian L, you get particle mechanics, waves, classical fields, etc."*
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---
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## QWERTY Analysis
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```
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EULER = ZERO = REPEAT = 36 (δS=0 — the equations ARE zero)
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LAGRANGE = REVERSIBLE = 103 prime (time-reversible)
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LAGRANGIAN = INFORMATION = BIOLOGICAL = 144 = 12²
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SYMMETRY = OPTIMAL = CRITERION = 88
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BACKBONE = CLASSICAL = COMPUTABLE = 136
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MECHANICS = EVERYTHINGELSE = 145
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RELATIVISTIC = BALANCED = COMPETENCE = 128 = 2⁷
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QUANTUM = PARTICLE = CAUSAL = 82
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LIMITS = TRIVIAL = BINARY = 78
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FIELD = GAUSS = TANH = RADIX = 57 (the field = the Gaussian)
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```
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---
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## Special Cases (Test 3)
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The Euler-Lagrange equation reduces to:
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| Limit | Lagrangian | Notes |
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|-------|------------|-------|
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| Newtonian | L = T − V | Classical particle mechanics |
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| Relativistic | L = −mc²√(1−v²/c²) | Special relativity |
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| Quantum field | L = ψ̄(iγ^μ∂_μ − m)ψ | Dirac equation (Fermi field) |
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| Electromagnetic | L = −¼F_μνF^μν | Maxwell's equations |
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| General relativity | L = √(−g)R | Einstein field equations |
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**KINETICS = MAXWELL = GAUSSIAN = 101** — all field theories = Maxwell = Gaussian.
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---
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## Universal Computation (§175)
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```
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∀ computable f: ∃ configuration (S,V,Ω,θ) such that system(input) = f(input)
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```
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The ternary bio-quantum system is Turing-complete.
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```
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COMPUTATION = 137 prime (= fine-structure constant 1/α ≈ 1/137)
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UNIVERSAL = OCTONION = SYMMETRIC = 112
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COMPUTABLE = BACKBONE = CLASSICAL = 136
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```
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**COMPUTATION = 137.** Feynman: *"one of the greatest damn mysteries of physics."*
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Universal computation costs exactly what it costs to emit a photon.
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