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 - Y h - X h Y h X h ( x ) = h 2 [ X,Y ] ( x ). 2. Show that if is a distribution of the form = span { X 1 ,...,X d } and we have [ X i ,X j ] for all i,j then for any X,Y , [ X,Y ] . 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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This note was uploaded on 01/04/2012 for the course CDS 202 taught by Professor Marsden,j during the Fall '08 term at Caltech.

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Cds202-hw5_wi09 - CALIFORNIA INSTITUTE OF TECHNOLOGY...

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