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Unformatted text preview: SYMPLECTIC GEOMETRY, LECTURE 4 Prof. Denis Auroux 1. Hamiltonian Vector Fields Recall from last time that, for ( M, ) a symplectic manifold, H : M R a C function, there exists a vector field X H s.t. i X H = dH . Furthermore, the associated ow t of this vector field is an isotopy of symplectomorphisms. Example. Consider S 2 R 3 with cylindrical coordinates ( r, , z ) and symplectic form = d dz ( is the usual area form). Then setting H = z gives the vector field : the associated ow is precisely rotation by angle t . Note also that the critical points of H are the fixed points of t , and t preserves the level sets of H , i.e. (1) dt d ( H t ) = dt d ( t H ) = t ( L X H H ) = t ( i X H ( X H )) = t ( ( X H , X H )) = One can apply this to obtain the ordinary formula for conservation of energy. Definition 1. X is a symplectic vector field if L X = 0 , i.e. i X is closed. X is Hamiltonian if i X is exact. By Poincar e, we see that, locally, symplectic vector fields are Hamiltonian. Globally, we obtain a class [ i X ] H 1 ( M, R )....
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This note was uploaded on 03/08/2010 for the course MATHEMATIC 570 taught by Professor Denisauroux during the Spring '10 term at Caltech.
- Spring '10