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Instituto Superior Tecnico Departamento de Matematica
 

Summary: Instituto Superior T´ecnico
Departamento de Matem´atica
SYMPLECTIC GEOMETRY - 2nd
Semester 2010/11
Problem Set 3
Due date: April 15
1. Let (M, ) be a closed symplectic manifold, f, g C
(M) two smooth functions
and t, t Ham(M, ) the (autonomous) Hamiltonian flows generated by Xf and
Xg. Show that t t = t t , t [0, 1], if and only of the Poisson bracket
{f, g} (Xf , Xg) C
(M) is identically zero.
2. Show that the flux : Diff0(M, ) H1
(M, R), defined by
flux(t) =
1
0
[Xt ] dt ,
where t Diff0(M, ) satisfies
d

  

Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa

 

Collections: Mathematics