https://t.inks.page/wiki/Geodesic Follow any website on Inks inks.page https://en.wikiquote.org/wiki/Geodesic?t=2026-04-07 We begin by recalling that geodesics can be obtained as solutions of the Euler-Lagrange equation of a Lagrangian given by the kinetic energy. We define symplectic and contact manifolds and we set up the basic geometry of the tangent bundle; we introduce the connection map, horizontal and vertical subbundles, the Sasaki metric, the symplectic form and the contact form. We describe the main properties of these objects and we show that the geodesic flow is a Hamiltonian flow. Also, when we restrict the geodesic flow to the unit sphere bundle of the manifold, we obtain a contact flow. The contact form naturally induces a probability measure that is invariant under the geodesic flow and is called the Liouville measure.

• Gabriel P. Paternain, Geodesic Flows. Springer. 6 December 2012. ISBN 9781461216001.  (1st edition, 1999)

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