dg2test1 - Differential Geometry 2 Test 1 Let (M, ω ) be...

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Unformatted text preview: Differential Geometry 2 Test 1 Let (M, ω ) be a symplectic manifold and let H ∈ C (M ) be a smooth function. Say that F ∈ C (M ) is a constant of the H -motion if and only if F is constant along each integral curve of the Hamiltonian vector field ξH . (a) Prove that F is a constant of the H -motion if and only if the Poisson bracket {F, H } is zero. (b) Prove that the set of all constants of the H -motion is a subspace of C (M ) that is closed under Poisson bracket. (c) What conclusion can be drawn from the hypothesis that every smooth function on M is a constant of the H -motion? (d) Can ξH have a dense integral curve? Solutions (a) Let γ be an integral curve of ξH and compute: (F ◦ γ ) (t) = γ (t)F = ξH (γ (t))F = ξH F (γ (t)) = {F, H }(γ (t)) ˙ so that F is constant along γ iff {F, H } is zero along γ . (b) Let CH (M ) denote the set of all constants of the H -motion. That CH (M ) is a subspace of C (M ) follows immediately from the fact that {·, ·} is real-linear in its first variable. That CH (M ) is closed under Poisson bracket follows from the Jacobi identity: if F1 , F2 ∈ C (M ) then {{F1 , F2 }, H } + {{F2 , H }, F1 } + {{H, F1 }, F2 } = 0; if F1 , F2 ∈ CH (M ) then {F2 , H } = {H, F1 } = 0 and so {{F1 , F2 }, H } = 0. (c) If every F ∈ C (M ) satisfies ξH (F ) = 0 then ξH = 0 so that dH = −ξH ω = 0. It follows that H is locally constant; in particular, if M is connected then H is constant. (d) Recall that H is constant along each integral curve of ξH . If (the image of) some integral curve is dense, it follows that H is constant on a dense set and hence (by continuity) constant; this forces ξH to be zero and therefore to have constant (singleton image) integral curves. Absurd. 1 ...
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