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Cds202-hw5_wi09

Cds202-hw5_wi09 - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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Unformatted text preview: CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #5 Issued: 6 Feb 09 (Fri) Due: 13 Feb 09 (Fri) Reading: Abraham, Marsden, and Ratiu (MTA), section 4.2, 4.4 Problems: 1. Consider the following vector fields on R 3 : X ( x ) = ∂ ∂x 2 − x 1 ∂ ∂x 3 Y ( x ) = ∂ ∂x 1 . Let x = (0 , , 0). Show that φ- Y h ◦ φ- X h ◦ φ Y h ◦ φ X h ( x ) = h 2 φ [ X,Y ] ( x ). 2. Show that if Δ is a distribution of the form Δ = span { X 1 ,...,X d } and we have [ X i ,X j ] ∈ Δ for all i,j then for any X,Y ∈ Δ, [ X,Y ] ∈ Δ. That is, to check involutivity of a distribution, we need only check that the pairwise brackets between basis elements lie in the distribution. 3. [Boothby, page 164, #4] Let N ⊂ M be a submanifold and let X,Y ∈ X ( M ) be vector fields such that X p ,Y p ∈ T p N for p ∈ N . Show that [ X,Y ] p ∈ T p N for all p ∈ N ....
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Cds202-hw5_wi09 - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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