Local expressions for Poisson manifolds of geodesic arcs in Lagrangian mechanics

Lubomír Klapka

E-mail: Lubomir.Klapka@math.slu.cz

Abstract. The local expressions for general Poisson manifolds of geodesic arcs are rather complicated. This paper shows that they can be simplified if the Poisson manifold is generated by some Lagrange function. Further, a second simplification is found by changing contravariant velocities to covariant velocities. Finally, local expressions for the corresponding Hamilton function and the corresponding linear connection are given.

AMSclassification. 17B65, 53B05, 53C15, 53C22, 58D15, 58F05, 70H99

Keywords. Geodesic arcs, geodesics, Lagrangian mechanics, linear connections, manifolds of geodesic arcs, Poisson algebras, Poisson manifolds, symplectic geometry