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**Extra resources for 4th Geometry Festival, Budapest**

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But pr1 ◦ γ is the identity map on M1 and pr2 ◦ γ = ϕ. Therefore, γ∗ω = 0 ⇐⇒ ϕ ∗ ω2 = ω 1 . Lecture 3 Generating Functions Generating functions provide a method for producing symplectomorphisms via lagrangian submanifolds. We will illustrate their use in riemannian geometry and dynamics. We conclude with an application to the study of the group of symplectomorphisms and to the problem of the existence of fixed points, whose first instance is the Poincar´e-Birkhoff theorem. 1 Constructing Symplectomorphisms Let X1 , X2 be n-dimensional manifolds, with cotangent bundles M1 = T ∗ X1 , M2 = T ∗ X2 , tautological 1-forms α1 , α2 , and canonical 2-forms ω1 , ω2 .

Exercise 20 Show that there exists a number c such that for 0 < h < c the level curve H = h in the (θ, ξ) plane is a disjoint union of closed curves. Show that the projection of each of these curves onto the θ-axis is an interval of length less than π. Show that neither of these assertions is true if h > c. What types of motion are described by these two types of curves? What about the case H = c? 2 The name “angle coordinates” is used even if the fibers are not tori. 56 LECTURE 4. HAMILTONIAN FIELDS Modulo 2π in θ, the function H has exactly two critical points: a critical point s where H vanishes, and a critical point u where H equals c.

Hint: The tangent space to the orbit being generated by the vector fields X # , this is a consequence of the Jacobi identity in g. This canonical symplectic form on the coadjoint orbits is also known as the Lie-Poisson or Kostant-Kirillov symplectic structure. (f) The Lie algebra structure of g defines a canonical Poisson structure on g∗ : {f, g}(ξ) := ξ, [dfξ , dgξ ] for f, g ∈ C ∞ (g∗ ) and ξ ∈ g∗ . Notice that dfξ : Tξ g∗ identified with an element of g g∗∗ . Check that {·, ·} satisfies the Leibniz rule: g∗ → R is {f, gh} = g{f, h} + h{f, g} .