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By Virgil Snyder, Charles Herschel Sisam

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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. 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.

10. The cotangent bundle T ∗ M \ {0} of manifold M is a symplectic manifold with a symplectic form ω given by ω = dθ = dxi ∧ dξi , where θ is the Poincar´e 1-form θ ∈ Ω1 T ∗ M \ {0} . ∂ i ∂ i ∂ Proof. It is clear that ω is closed. If X = a i ∂x i + b ∂ξ , and Y = v ∂xi + i wi ∂ξ∂ i , then ω(X, Y ) = a · w − b · v. By setting v = w we obtain a = b, and by setting w = 0 we obtain a = b = 0. The next example shows that we can always formulate Hamilton’s equations on the cotangent bundle. This motivates the name for X H .

The other proof is similar. 37 38 References [AIM94] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and finsler spaces with applications in physics and biology, Fundamental Theories of Physics, Kluwer Academic Publishers, 1994. [Ana96] M. Anastasiei, Finsler Connections in Generalized Lagrange Spaces, Balkan Journal of Geometry and Its Applications 1 (1996), no. 1, 1–10. [Con93] L. 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.

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