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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 2
Due date: March 31
1. Let (M, ) be a symplectic manifold, : M M a symplectomorphism and h : M
R a smooth function. Show that the Hamiltonian vector fields Xh and Xh-1 satisfy
the relation
Xh-1 = Xh .
2. Let L be a smooth manifold and T
L its cotangent bundle equipped with the canonical
symplectic form can = -dcan. Let also h : L R be a smooth function and
h : T
L T
L a diffeomorphism defined by
h(x, ) = (x, + dhx) , x L , T
x L .
Show that

  

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

 

Collections: Mathematics