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ExercisesWk3 - CDS 140b Homework Set 2 Due by Thursday For...

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CDS 140b: Homework Set 2 Due by Thursday, January 31, 2008. For the purpose of calibration, note down the time spent on each problem on your solution. Problems 1. Consider the Poisson form of the rigid body equations. Show that the total angular momentum C ( Π ), defined as C ( Π ) = 1 2 k Π k 2 Poisson commutes with any function, i.e. for any function G ( Π ), { C, G } = 0 . A function Poisson commuting with any other function is called a Casimir function . 2. A vector field X acts as a derivation on functions as follows: X ( f ) = Df · X , i.e. X ( f ) is the derivative of f in the direction of X . (a) Show that if X F is a vector field on phase space associated to a function F ( q, p ), then X F ( f ) = { f, F } (1) for all functions f . (b) Show by direct calculation that for any two functions F and G on phase space the following relation holds: X { F,G } = - [ X F , X G ] (2) where the Poisson bracket is the canonical one on phase space, and the Lie bracket [ X, Y ] of two vector fields is defined by [ X, Y ]( f ) = X ( Y ( f )) - Y ( X ( f )) . (3) (c) Now consider an arbitrary Poisson structure on (a subset of)
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