ExercisesWk4

# ExercisesWk4 - explicitly to show that H I 2 x I 2 y = 0...

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CDS 140b: Homework Set 3 Due by Thursday, February 7, 2008. Problems The Poisson bracket for the N -vortex problem is given by { f,g } = N X i =1 1 Γ i ± ∂f ∂x i ∂g ∂y i - ∂g ∂x i ∂f ∂y i ² . for arbitrary functions f ( x 1 ,y 1 ; ... ; x N ,y N ) and g ( x 1 ,y 1 ; ... ; x N ,y N ) on R 2 N . 1. Show that the following relations hold: { H, L} = 0 , { H, I x } = 0 , { H, I y } = 0 , {I x , I y } = N X i =1 Γ i , {I x , L} = 2 I y , {I x , L} = - 2 I x , where H , L , I x and I y are the Kirchhoﬀ-Routh function, the angular impulse, and the x and the y component of the linear impulse, respectively. See the lecture notes for explicit expressions. 2. Use the general properties of the Poisson bracket ( i.e. do not calculate
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Unformatted text preview: explicitly) to show that { H, I 2 x + I 2 y } = 0 and {L , I 2 x + I 2 y } = 0 , i.e. H , L , and I 2 x + I 2 y are three (independent) integrals of the motion whose Poisson bracket vanishes. Project ideas 1. Numerically explore chaos in the four-vortex problem ( i.e. draw Poincar e maps, (maybe) Melnikovs method, . ..). 2. Look at the existence of relative equilibria in the N-vortex problem. 3. Compute LCS for problems in chaotic advection. 1...
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