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  "body": "\n## Example\n\nLet $M$ be a complete [[Riemannian manifold|5HDG31YEPW]] with [[tangent bundle|DQB1BVQJWD]] $TM$.\nLet $SM$ be its unit tangent bundle, the set of tangent vectors of length $1$.\nFor $v \\in SM$ let $\\gamma_v$ be the [[geodesic|NSYMQMHPDN]] with $\\gamma_v(0)$ the base point of $v$ and $\\gamma_v'(0) = v$.\nThe geodesic flow is $\\varphi_t(v) = \\gamma_v'(t)$ for $t \\in \\R$.\nIt is defined for all $t$ because $M$ is complete, and $\\varphi_{s+t} = \\varphi_s \\circ \\varphi_t$ because $t \\mapsto \\gamma_v(s + t)$ is the geodesic with initial vector $\\gamma_v'(s)$.\nSo it is a [[smooth dynamical system|116TPXR5FS]] in continuous time on $SM$.\n\nCheck on the flat torus $M = \\R^2/\\Z^2$: geodesics are straight lines, so $SM = M \\times S^1$ and $\\varphi_t(x, v) = (x + t v, v)$.\nFor a direction $v$ of irrational slope, the orbit of $(x, v)$ is dense in the torus $M \\times \\set{v}$.\nFor a direction of rational slope, every orbit is periodic.\n\n## Remarks\n\nThe geodesic flow preserves the Liouville measure on $SM$, the product of the Riemannian volume and the uniform measure on each unit sphere.\nIf $M$ is compact and its [[sectional curvature|1X42JAA349]] is negative everywhere, the geodesic flow is an [[Anosov flow|8FQJXSA4DT]], as Anosov proved, and it is ergodic for the Liouville measure.\nFor a surface of constant curvature $-1$ it can be written as a homogeneous flow, see [[geodesic flow on a locally symmetric space|5TKBSHY6VC]].\nIdentify $TM$ with the cotangent bundle $T^*M$ by the metric.\nThen the geodesic flow is the Hamiltonian flow of the kinetic energy $H(v) = \\frac12\\norm{v}^2$, restricted to an energy level.\n\n## Explanations\n\n### L1 human\n\nPut a frictionless marble on a curved surface and push it in some direction.\nIt rolls along the straightest possible path, a geodesic, at constant speed.\nThe geodesic flow records where the marble is and which way it moves after time $t$.\nOn a flat torus the motion is simple, but on a surface curved like a saddle everywhere, nearby paths diverge exponentially and the motion is chaotic.\n",
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