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Chapter11-page16 - Physics 235 Chapter 11 1 = $ cos sin 2 =...

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Physics 235 Chapter 11 - 16 - ! " 1 ! # = ! $ cos # sin % ! " 1 ! ! # = cos % ! " 2 ! # = ! $ cos # cos % ! " 2 ! ! # = & sin % ! " 3 ! # = & ! $ sin # ! " 3 ! ! # = 0 and Lagrange's equation becomes ! ! I 1 " 1 sin # + I 2 " 2 cos # { } cos $ % I 3 " 3 sin $ ( ) % d dt I 1 " 1 cos # % I 2 " 2 sin # { } = 0 The Euler angle ψ : Lagrange's equation for the coordinate ψ is 0 = ! T ! " # d dt ! T ! ! " = ! T ! $ i ! $ i ! " i % # d dt ! T ! $ i ! $ i ! ! " i % = I i $ i ! $ i ! " i % # d dt I i $ i ! $ i ! ! " i % = 0 Differentiating the angular velocity with respect to the coordinate ψ we find ! " 1 ! # = ! $ sin % cos # & ! % sin # = " 2 ! " 1 ! ! % = 0 ! " 2 ! # = & ! $ sin % sin # & ! % cos # = & " 1 ! " 2 ! ! % = 0 ! " 3 ! # = 0 ! " 3 ! ! % = 1 and Lagrange's equation becomes I 1 ! 1 ! 2 " I 2 ! 2 ! 1 " d dt I 3 ! 3 { } = I 1 " I 2 ( ) ! 1 ! 2 " d dt I 3 ! 3 { } = I 1 " I 2 ( ) ! 1 ! 2 " I 3 ! ! 3 = 0 Of all three equations of motion, the last one is the only one to contain just the components of the
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