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**Sample text**

In consequence, D(Φ−1 ◦ c) = DΦ−1 ◦ Xh ◦ c = (Φ∗ Xh )(Φ−1 ◦ c) = Xh◦Φ (Φ−1 ◦ c), so Φ−1 ◦ c is an integral curve of Xh◦Φ . 1 Symplectic structure on T ∗M \ {0} For any manifold its cotangent bundle is a symplectic manifold. 8 (Poincar´e 1-form). Suppose M is an manifold. Then the Poincar´e 1-form θ ∈ Ω1 T ∗ M \ {0} is defined as θ = −ξi dxi . where (xi , ξi ) are local coordinates for T ∗ M \ {0}. If (˜ xi , ξ˜i ) are other standard coordinates for T ∗ M \{0}, then ξi = r i and ξi dxi = ∂∂xx˜ i ξ˜r ∂x d˜ xl = ξ˜i d˜ xi .

5 (Symplectic mapping). Suppose (M, ω) and (N, η) are symplectic manifolds of the same dimension, and f is a diffeomorphism Φ : M → N . Then Φ is a symplectic mapping if Φ ∗ η = ω. 6. Suppose (M, ω) is a symplectic manifold, and X H is a Hamiltonian vector field corresponding to a function H : M → R. Furthermore, suppose Φ : I × U → M is the local flow of X H defined in some open 31 U ⊂ M and open interval I containing 0. Then for all x ∈ U , t ∈ I, we have (Φ∗t ω)x = ωx , where Φt = Φ(t, ·). Proof.

F is a constant on integral curves of G and G/F . Proof. If c is in integral curve of G, and L is the symplectic mapping induced by F , then L ◦ c is an integral curve of X 1 F 2 ◦L −1 . 4 2 implies that F 2 ◦ L −1 ◦ L ◦ c is constant. The proof of the second claim is analogous. 9. 16. A stationary curve of E is a geodesic. Proof. Suppose E is stationary for a curve c. 6 implies that cˆ is a integral curve of G. 4. 8. 17. If γ : I → T M \ {0} is an integral curve of G/F , then π ◦ γ is a stationary curve for E.