A brief introduction to Finsler geometry by Dahl M.

By Dahl M.

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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 . Hence θ is well defined. 9 (Coordinate independent expression for θ). Let π be the canonical projection π : T ∗ M → M . Then the Poincar´e 1-form θ ∈ Ω1 (T ∗ M ) satisfies θξ (v) = ξ (Dπ)(v) for ξ ∈ T ∗ Q and v ∈ Tξ T ∗ Q . Proof. Let (xi , yi ) be standard coordinates for T ∗ Q near ξ.

Since c is an integral curve, we have (Dc)(t, 1) = (X H ◦ c)(t), so for τ = (t, 1) ∈ Tt I, we have d(H ◦ c)t (τ ) = (c∗ dH)t (τ ) = (dH)c(t) (Dc)(τ ) = (dH)c(t) (XH ◦ c)(t) = 0, since ω is antisymmetric. The claim follows since d(H ◦ c) is linear. 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.

30 7 Symplectic geometry Next we show that T ∗ M \ {0} and T M \ {0} are symplectic manifolds, and study geodesics and the Legendre transformation in this symplectic setting. 1. Suppose ω is a 2-form on a manifold M . Then ω is nondegenerate, if for each x ∈ M , we have the implication: If a ∈ T x M , and ωx (a, b) = 0 for all b ∈ Tx M , then a = 0. 2 (Symplectic manifold). Let M be an even dimensional manifold, and let ω be a closed non-degenerate 2-form on M . Then (M, ω) is a symplectic manifold, and ω is a symplectic form for M .

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