Symplectic geometryLecture 6Symplectic manifolds.Examples.Hamiltonian mechanics.Definition of a symplectic manifold.Symplectomorphisms.Symplectic and Hamiltonian vector fields.These are called symplectic vector fields.Hamiltonian vector fields are symplectic.The bracket of two symplectic vector fields is Hamiltonian.Poisson brackets.Poisson algebras.Siméon Denis PoissonBorn: 21 June 1781 in Pithiviers, FranceDied: 25 April 1840 in Sceaux (near Paris), FrancePoisson manifolds.The basic local example.Hamilton’s equations.Quadratic Hamiltonians.Quadratic Hamiltonians - mixed type.Polynomials in p .TextPolynomials in q .The cotangent bundle.Galileo.F=maHorizontal Lagrangian submanifolds of the cotangent bundle..Exact symplectic manifolds.The effect of the “magnetic
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