Συμπλεκτική Γεωμετρία και Χαμιλτονιανή Μηχανική

Symplectic Geometry is a branch of Differential Geometry and it studies the properties of symplectic manifolds. A symplectic manifold is a differentiable manifold equipped with a non- degenerate closed 2-form. Symplectic manifolds are the natural place to do Hamiltonian mechanics since the phase space of a mechanical system is the cotangent bundle of the configuration space. Any smooth real valued function H, called Hamiltonian, can induce a vector field on the symplectic manifold. This vector field is called Hamiltonian vector field. The integral lines of this vector field represent solutions of Hamilton's equations and the diffeomorphisms arising from the flow of the vector field are called symplectomorphisms. Those symplectomorphisms are known in Physics as canonical transformations, changes of coordinates that preserve the form of Hamilton's equations. The symplectic form is preserved by the Hamiltonian flow (Liouville's theorem). Symplectic Geometry is the geometry of phase space, the geometric background for Hamiltonian Mechanics. Our purpose was to present an introduction to Symplectic Geometry and its notions and to state basic theorems of Classical Mechanics like Liouville's, Noether' s and Arnold-Liouville's theorems in symplectic language