Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 337H 2 (M, R) is a necessary condition for the compact manifold to admit asymplectic structure.However, if M is a 2nD compact manifold without boundary, then theredoes not exist any exact symplectic structure, ω = dθ on M, as its totalvolume is zero (by Stokes’ Theorem),∫MΩ ω =n(n−1)(−1) 2n!∫Mω n =n(n−1)(−1) 2n!∫Md(θ ∧ ω n−1 ) = 0.For example, spheres S 2n do not admit a symplectic structure for n ≥2, since the second de Rham group vanishes, i.e., H 2 (S 2n , R) = 0. Thisargument applies to any compact manifold without boundary and havingH 2 (M, R) = 0.In mechanics, the phase–space is the cotangent bundle T ∗ M of a configurationspace M. There is a natural symplectic structure on T ∗ M thatis usually defined as follows. Let M be a smooth nD manifold and picklocal coordinates {dq 1 , ..., dq n }. Then {dq 1 , ..., dq n } defines a basis of thetangent space T ∗ q M, and by writing θ ∈ T ∗ q M as θ = p i dq i we get localcoordinates {q 1 , ..., q n , p 1 , ..., p n } on T ∗ M. Define the canonical symplecticform ω on T ∗ M byω = dp i ∧ dq i .This 2−form ω is obviously independent of the choice of coordinates{q 1 , ..., q n } and independent of the base point {q 1 , ..., q n , p 1 , ..., p n } ∈ Tq ∗ M;therefore, it is locally constant, and so dω = 0.The canonical 1−form θ on T ∗ M is the unique 1−form with the propertythat, for any 1−form β which is a section of T ∗ M we have β ∗ θ = θ.Let f : M → M be a diffeomorphism. Then T ∗ f preserves the canonical1−form θ on T ∗ M, i.e., (T ∗ f) ∗ θ = θ. Thus T ∗ f is symplectic diffeomorphism.If (M, ω) is a 2nD symplectic manifold then about each point x ∈ Mthere are local coordinates {q 1 , ..., q n , p 1 , ..., p n } such that ω = dp i ∧ dq i .These coordinates are called canonical or symplectic. By the DarbouxTheorem, ω is constant in this local chart, i.e., dω = 0.