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8. 17. 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.

Equation (45) follows from the definition, equation (46) follows from m Rjk ∂Nkm = − Njs Gm sk − ∂xj ∂Njm − Nks Gm sj , ∂xk and equation (44) follows from equation (46). 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.

Conlon, Differentiable manifolds: A first course, Birkh¨auser, 1993. N. Dzhafarov and H. Colonius, Multidimensional fechnerian scaling: Basics, Journal of Mathematical Psychology 45 (2001), no. 5, 670–719. S. Ingarden, On physical applications of finsler geometry, Contemporary Mathematics 196 (1996). [Kap01] E. Kappos, Natural metrics on tangent bundle, Master’s thesis, Lund University, 2001. [KT03] L. Kozma and L. Tam´assy, Finsler geometry without line elements faced to applications, Reports on Mathematical Physics 51 (2003).