#theoreticalscience — Public Fediverse posts
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https://openai.com/index/ten-advances-in-mathematics/ #groundbreakingmath #techhumor #browsersettings #theoreticalscience #HackerNews #ngated -
Noether’s Theorem: Symmetry→Conservation Law.
Symmetries of the action imply conserved quantities per Noether's theorem.
A path q(t) is shifted by δq = φ but smoothed in transition layers τ wide at t0 and t1, where Δq̇ ≈ φ/τ. Action changes there approximate ΔS ≈ (∂L/∂q̇) Δq̇ τ ≈ −(∂L/∂q̇ φ)(t1) and ≈ +(∂L/∂q̇ φ)(t0).
Symmetry sets total ΔS = 0, hence (∂L/∂q̇ φ) is conserved from t0 to t1.Momentum conservation's derivation from space translation symmetry and energy conservation from time translation symmetry in mechanics and physics is done by it.
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