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(d) Cyclic coordinates? ? θ appears explicitly in ? L through cos ⁡ ? cosθ, so ? θ is not cyclic. No simple momentum conservation beyond energy (the Lagrangian has no explicit time dependence, so the energy ? = ? + ? E=T+V is conserved). (e) Equilibria: set sin ⁡ ? = 0 sinθ=0 → ? = 0 θ=0 (stable, pendulum down) and ? = ? θ=π (unstable, inverted). Small oscillations about ? = 0 θ=0: linearize sin ⁡ ? ≈ ? sinθ≈θ

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wa2115

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