By John Playfair
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Extra info for Elements of Geometry: Containing the first six books of Euclid, with a supplement on the quadrature of the circle, and the geometry of solids; to which are added, Elements of plane and spherical trigonometry
If c : I → M is a stationary curve for E, then λ = F ◦ cˆ is constant and L −1 ◦ c ◦ M1/λ is an integral curve of Xh . 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.
What is more, XF = G/F . 2 Proof. Using dη = − ∂gij i r y dx ∧ dxj + gij dxi ∧ dy j , ∂xr we obtain dη(G, ·) = = ∂gij ∂gis 1 ∂F 2 i i j i s − dy y y + 2g G dx + is ∂xs ∂xj 2 ∂y i 1 ∂F 2 i 1 ∂F 2 i dx + dy . 2 ∂xi 2 ∂y i The second claim follows since ιXF ω = dF = ιG/F ω. 14 state that dη is preserved under the flow of G. 15. F is a constant on integral curves of G and G/F . Proof. If c is in integral curve of G, and L is the symplectic mapping induced by F , then L ◦ c is an integral curve of X 1 F 2 ◦L −1 .
Suppose M, N are manifolds, Ψ : M → N is a diffeomorphism. Then the pullback of Ψ for vector fields is the mapping Ψ∗ : X (N ) → X (M ) , Y → (DΨ−1 ) ◦ Y ◦ Ψ. 7. Suppose (M, ω), (N, η) are symplectic manifolds, Φ : M → N is a symplectic mapping such that Φ ∗ η = ω, and h : N → R is a smooth function. Then Φ∗ (Xh ) = Xh◦Φ . What is more, if c : I → N is an integral curve of X h ∈ X (N ), then Φ−1 ◦ c is an integral curve of Xh◦Φ . Proof. The contraction operator satisfies ιΦ∗ X (Φ∗ η) = Φ∗ (ιX η) for all η ∈ Ωk (N ), X ∈ X (N ).
Elements of Geometry: Containing the first six books of Euclid, with a supplement on the quadrature of the circle, and the geometry of solids; to which are added, Elements of plane and spherical trigonometry by John Playfair