Transcript Physics 321 Hour 32 Lagrangian Treatment of Rigid Bodies

Physics 321
Hour 32
Lagrangian Treatment of Rigid Bodies
Football in Empty Space
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𝑇 = 𝐼11 𝜑 sin 𝜃 + 𝜃 + 𝐼33 ψ + 𝜑 cos 𝜃
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ℒ =𝑇−𝑈 =𝑇
𝜑 and ψ are both ignorable, so the corresponding
angular momenta are conserved.
𝜕ℒ
𝑝ψ =
= 𝐼33 ψ + 𝜑 cos 𝜃 = 𝐿3
𝜕ψ
𝑝𝜑 = 𝐼33 ψ + 𝜑 cos 𝜃 cos 𝜃 + 𝐼11 𝜑sin2 𝜃 = 𝐿𝑧
𝐿𝑧 − 𝐿3 cos 𝜃
𝜑=
𝐼11 sin2 𝜃
Football in Empty Space
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𝑇 = 𝐼11 𝜑 2 sin2 𝜃 + 𝜃 2 + 𝐼33 ψ + 𝜑 cos 𝜃
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ℒ =𝑇−𝑈 =𝑇
See T football.nb and Lagrange football.nb
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Top Lagrangian
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ℒ = 𝐼11 𝜑 2 sin2 𝜃 + 𝜃 2 + 𝐼33 ψ + 𝜑 cos 𝜃
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−𝑀𝑔𝑅 cos 𝜃
𝜑 and ψ are still ignorable.
See Lagrangian top.nb
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