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lect04 - SYMPLECTIC GEOMETRY LECTURE 4 Prof Denis Auroux 1...

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SYMPLECTIC GEOMETRY, LECTURE 4 Prof. Denis Auroux 1. Hamiltonian Vector Fields Recall from last time that, for ( M, ω ) a symplectic manifold, H : M R a C function, there exists a vector field X H s.t. i X H ω = dH . Furthermore, the associated flow ρ t of this vector field is an isotopy of symplectomorphisms. Example. Consider S 2 R 3 with cylindrical coordinates ( r, θ, z ) and symplectic form ω = dz ( ω is the usual area form). Then setting H = z gives the vector field : the associated flow is precisely rotation by angle ∂θ t . Note also that the critical points of H are the fixed points of ρ t , and ρ t preserves the level sets of H , i.e. (1) dt d ( H ρ t ) = dt d ( ρ t H ) = ρ t ( L X H H ) = ρ t ( i X H ω ( X H )) = ρ t ( ω ( X H , X H )) = 0 One can apply this to obtain the ordinary formula for conservation of energy. Definition 1. X is a symplectic vector field if L X ω = 0 , i.e. i X ω is closed. X is Hamiltonian if i X ω is exact. By Poincar´ e, we see that, locally, symplectic vector fields are Hamiltonian. Globally, we obtain a class [ i X ω ] H 1 ( M, R ). Example. On T 2 , ∂x and ∂y are symplectic vector fields: since dy and dx are not exact, ∂x and ∂y are not Hamiltonian.
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