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Proof. In the first claim, L ◦ γ is an integral curve of X F . 17. 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.

Then dci dt dpi dt ∂H ◦ γ, ∂ξi ∂H = − i ◦ γ, ∂x = that is, integral curves of XH are solutions to Hamilton’s equations in local coordinates of T ∗ M . 2 Symplectic structure on T M \ {0} The previous section shows that T ∗ M \ {0} is always a symplectic manifold. No such canonical symplectic structure is known for the tangent bundle. However, if M is a Finsler manifold, then T M \ {0} has a canonical symplectic structure induced by the Hilbert 1-form on T M \ {0}. 12 (Hilbert 1-form). Let F be a Finsler norm on M .

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). [MA94] R. Miron and M. 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.

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