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ExercisesWk6

# ExercisesWk6 - 1 CDS 140b Winter 2008 Week 6 Spherical...

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1 CDS 140b Winter 2008 Week 6 Spherical Tippe Top Inversion as a Dissipation-Induced Instability Due: Thursday, Feb 21, 2008 This assignment references the preprint “Dissipation-Induced Heteroclinic Orbits in Tippe Tops” which accompanies this homework. Problem A [Quadratic Lagrangians] Let M L ( R n , R n ) be a symmetric positive-definite matrix. Consider a me- chanical system with quadratic Lagrangian L : R 2 n R : L ( q , ˙ q ) = ˙ q T M ˙ q + ˙ q T G q - q T K q . Show that the only conservative forces derivable from such a quadratic La- grangian are gyroscopic and potential. (Recall a gyroscopic force is a force that can be written as A ˙ q where A is skew-symmetric and a potential force is a force that can be written as B q where B is symmetric.) Problem B [Linear Hamiltonian Systems] Let q ( t ) R 2 . Consider A, B L ( R 2 , R 2 ) where A is skew-symmetric and B is symmetric. Verify that the following differential equations are Hamiltonian: ¨ q + A ˙ q + B q = 0. (*) Show that if σ is an eigenvalue of (*) then so are 1 , ¯

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ExercisesWk6 - 1 CDS 140b Winter 2008 Week 6 Spherical...

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