Geodesic
A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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We review properties of so-called special conformal Killing tensors on a Riemannian manifold $(Q,g)$ and the way they give rise to a Poisson–Nijenhuis structure on the tangent bundle $TQ$. We then address the question of generalizing this concept to a Finsler space, where the metric tensor field comes from a regular Lagrangian function $E$, homogeneous of degree two in the fibre coordinates on $TQ$. It is shown that when a symmetric type (1,1) tensor field $K$ along the tangent bundle projection $\tau\colon TQ\rightarrow Q$ satisfies a differential condition which is similar to the defining relation of special conformal Killing tensors, there exists a direct recursive scheme again for first integrals of the geodesic spray. Involutivity of such integrals, unfortunately, remains an open problem.
Keywords: special conformal Killing tensors; Finsler spaces. 
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     author = {Willy Sarlet},
     title = {A~Recursive {Scheme} of {First} {Integrals} of the {Geodesic} {Flow} of {a~Finsler} {Manifold}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a23/}
}
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Willy Sarlet. A Recursive Scheme of First Integrals of the Geodesic Flow of a Finsler Manifold. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a23/

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