Cds202-hw6_wi09

Cds202-hw6_wi09 - X ⊂ TG is a left invariant vector...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 202 R. Murray Winter 2009 Problem Set #6 Issued: 12 Feb 09 Due: 19 Feb 09 Reading: Abraham, Marsden, and Ratiu (MTA), Sections 5.1 and 5.2 Problems: 1. MTA 5.1-1: properties of the adjoint map on GL( n ) 2. MTA 5.1-2: tangent group 3. [Warner, page 135, #16; MTA 5.1-5] (a) Let G be a Lie group. Show that the set of right invariant vector ﬁelds on G forms a Lie algebra under the Lie bracket operation and that it is naturally isomorphic to T e G . (b) Let φ : G G be the diﬀeomorphism deﬁned by φ ( g ) = g - 1 . Prove that if
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Unformatted text preview: X ⊂ TG is a left invariant vector ﬁeld on G then φ * ( X ) is a right invariant vector ﬁeld whose value at e is-X ( e ). Further show that X 7→ φ * ( X ) gives a Lie algebra isomorphism of the Lie algebra of left invariant vector ﬁelds with the Lie algebra of right invariant vector ﬁelds on G . (A Lie algebra isomorphism is a linear mapping A : V → V which preserves the Lie bracket: A [ ξ,η ] = [ Aξ,Aη ].) 4. MTA 5.2-1, parts (iii)–(v): calculations on SO(3) 5. MTA 5.2-5: the Euclidean group...
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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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