Cds202-hw4_wi09

# Cds202-hw4_wi09 - and show that it is φ invariant 6 Let...

This preview shows page 1. Sign up to view the full content.

CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #4 Issued: 30 Jan 09 (Fri) Due: 6 Feb 09 (Fri) Reading: Abraham, Marsden, and Ratiu (MTA), sections 3.3, 4.1, 4.2 Problems: 1. MTA 3.3-1: tangent spaces/maps for graphs 2. If F ( x 1 ,x 2 ,x 3 ) = 0 is a submersion deﬁning a 2-dimensional manifold in R 3 , under what conditions is X = v 1 ∂x 1 + v 2 ∂x 2 + v 3 ∂x ∂x 3 (evaluated at a point where F ( x 1 ,x 2 ,x 3 ) = 0) a tangent vector to M ? 3. [Boothby, page 119, #12] Show that any smooth vector ﬁeld Y on S n - 1 R n is the restriction of a smooth vector ﬁeld X on R n . 4. MTA 4.1-5: convolution equation 5. [Boothby, page 126, #6] Show that φ t ( x,y ) deﬁned by φ t ( x,y ) = ( xe 2 t ,ye - 3 t ) deﬁnes a C ﬂow on M = R 2 . Determine the vector ﬁeld that generates this ﬂow (called the inﬁnitesimal generator of the ﬂow
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) and show that it is φ invariant. 6. Let SO (3) be the set of 3 × 3 orthogonal matrices with determinant +1. The tangent space of SO (3) at the identity is given by the set of skew-symmetric matrices of the form b ω = ( ω ) ∧ = ±-ω 3 ω 2 ω 3-ω 1-ω 2 ω 1 ² (we’ll show this later in the course). (a) Show that if v ∈ R 3 , b ωv = ω × v , where × is the cross product in R 3 . (b) Show that the tangent space T R SO (3) consists of matrices of the form b ωR where b ω is skew-symmetric. (c) Show that the ﬂow of a vector ﬁeld g ( R ) = b ωR is given by φ t ( R ) = exp( b ωt ) R where exp is the matrix exponential....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online