homework1 - 18.966 – Homework 1 – due Thursday March 1...

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Unformatted text preview: 18.966 – Homework 1 – due Thursday March 1, 2007. 1. Show that, if E is a Lagrangian subspace of a symplectic vector space ( V, Ω), then any basis e 1 , . . . , e n of E can be extended to a standard symplectic basis e 1 , . . . , e n , f 1 , . . . , f n of ( V, Ω). 2. For which values of n does the sphere S 2 n ⊂ R 2 n +1 carry a symplectic structure? What about the torus T 2 n = R 2 n / Z 2 n = ( S 1 ) 2 n ? 3. Let { ρ t } t ∈ [0 , 1] be the isotopy generated by a time-dependent symplectic vector field X t dρ t on a symplectic manifold ( M, ω ), i.e. ρ 0 = Id, = X t ◦ ρ t , and i X t ω is closed. Then the dt flux of { ρ t } is defined to be 1 Flux( ρ t ) = [ i X t ω ] dt ∈ H 1 ( M, R ) . 0 a) Let γ : S 1 → M be an arbitrary closed loop, and define Γ : [0 , 1] × S 1 → M by the formula Γ( t, s ) = ρ t ( γ ( s )), so γ t ( · ) = Γ( t, · ) is the image of the loop γ by ρ t . Prove that ( Flux( ρ t ) , [ γ ] ) = Γ ∗ ω. (1) [0 , 1] × S 1 (Remark: the right-hand side is simply the symplectic area swept by the family of loops...
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homework1 - 18.966 – Homework 1 – due Thursday March 1...

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