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Anastasiei, The geometry of lagrange spaecs: Theory and applications, Kluwer Academic Press, 1994. [MS97] D. McDuff and D. Salamon, Introduction to symplectic topology, Clarendon Press, 1997. [Run59] H. Rund, The differential geometry of Finsler spaces, Springerverlag, 1959. [She01a] Z. Shen, Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, 2001. [She01b] , Lectures on Finsler geometry, World Scientific, 2001.

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 . 3 (Hamiltonian vector field). Suppose (M, ω) is a symplectic manifold, and suppose H be a function H : M → R. Then the Hamiltonian vector field induced by H is the unique (see next paragraph) vector field XH ∈ X (M ) determined by the condition dH = ι XH ω. In the above, ι is the contraction mapping ι X : Ωr M → Ωr−1 M defined by (ιX ω)(·) = ω(X, ·).

If γ : I → T M \ {0} is an integral curve of G/F , then π ◦ γ is a stationary curve for E. Conversely, if c is a stationary curve for E, then λ = F ◦ cˆ is constant and c ◦ M1/λ (see below) is an integral curve of G/F . If s > 0, we denote by Ms the mapping Ms : t → st, t ∈ R. 35 Proof. Let c : I → T M \ {0} be an integral curve of G/F . If c = (x, y), then dxi dt dy i dt = yi , λ = −2 Gi ◦ c , λ where λ = F ◦γ > 0 is constant. The first equation implies that c = λπ ◦ γ. Since Gi is 2-homogeneous, it follows that d2 xi + 2Gi (π ◦ c) = 0, dt2 so π ◦ c is a stationary curve for E.