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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les buildings permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de los angeles compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

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The next proposition shows how the Hamilton equations can be formulated using the Legendre transformation and Finsler quantities on the tangent bundle. 18. Suppose h : T ∗ M → R is a co-Finsler norm, L : T ∗ M → T M is the induced symplectic mapping, and G is the geodesic spray induced by the Finsler norm F = h ◦ L −1 . If γ is an integral curve of Xh , and λ = h ◦ γ > 0, then λπ ◦ γ = L ◦ γ. (47) A curve γ = (c, p) : I → T ∗ M \ {0} is an integral curve to Xh if and only if dci dt dpi dt = = 1 i y ◦ L ◦ γ, λ 1 m −1 (N L ) ◦ L ◦ γ.

Proof. Let (xi , yi ) be standard coordinates for T ∗ Q near ξ. Then we can ∂ i ∂ i ∂ write ξ = ξi dxi |π(ξ) and v = αi ∂x i |ξ +β ∂y |ξ , Thus (Dπ)(v) = α ∂xi |π(ξ) , i and θξ (v) = yi (ξ)dxi |ξ (v) = ξi αi = ξ (Dπ)(α) . 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.

If (xi , ξi ) are standard coordinates for T ∗ M \ {0}, then XH = ∂H ∂ ∂H ∂ − i . i ∂ξi ∂x ∂x ∂ξi Suppose that γ : I → T ∗ M \ {0} is an integral curve to XH and locally γ = (c, p). 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.

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A brief introduction to Finsler geometry by Dahl M.


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