View source for Geodesic flow

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A flow $\{S_t\}$ whose phase space is the manifold $TM^n$ of vectors tangent to a Riemannian (more generally, a Finsler) manifold $M^n$ (the so-called configuration manifold of the flow), while the motion is defined as follows. Let $v\in TM^n$ be a vector tangent to $M^n$ at a point $x\in M^n$ and let its length be $|v|\neq0$. Let a geodesic line $\gamma$ on $M^n$ be drawn through $x$ in the direction $v$ and let $x_t$ be the point on $\gamma$ the distance of which from $x$ along $\gamma$ is $t|v|$ (where that direction on $\gamma$ is considered to be positive which is identical with the direction of the vector $v$ at $x$). One then has $S_tv=v_t=dx_t/dt$. In case $|v|=0$, one has $S_tv\equiv v$. It turns out that $|v_t|=\text{const}$, and for this reason the vectors of unit length form a submanifold $W^{2n-1}$ in $TM^n$ that is invariant with respect to $\{S_t\}$. A geodesic flow is often understood to mean the restriction of the flow $\{S_t\}$ to $W^{2n-1}$. In local coordinates a geodesic flow is described by a system of ordinary second-order differential equations, which, in the Riemannian case, have the form

$$\frac{d^2x^i}{dt^2}+\sum_{j,k}\Gamma_{jk}^i(x_t)\frac{dx^j}{dt}\frac{dx^k}{dt}=0,$$

where $x^i$ is the $i$-th coordinate of the point $x_t$ and the $\Gamma_{jk}^i$ are the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) of the second kind. A geodesic flow preserves the natural [[Symplectic structure|symplectic structure]] on $TM^n$, while its restriction to $W^{2n-1}$ preserves the corresponding [[Contact structure|contact structure]]. Geodesic flows obviously play an important role in geometry (see also [[Variational calculus in the large|Variational calculus in the large]]). If, in addition, a certain change of time is made, then it is possible to reduce the description of the motion of a mechanical system, in accordance with the [[Maupertuis principle|Maupertuis principle]], to a geodesic flow.



====Comments====
For the application to mechanical systems, see, for example, Section 45D and Appendices 1J and 4F in [[#References|[a2]]]. The geodesic flows on (compact) manifolds of negative curvature have interesting dynamical properties (cf. [[Hyperbolic set|Hyperbolic set]]; [[Y-system|$Y$-system]]). See [[#References|[a1]]]. For applications of geodesic flows in differential geometry, see [[#References|[a3]]], Chapt. 3.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.V. Anosov,   "Geodesic flows on compact Riemannian manifolds of negative curvature"  ''Proc. Steklov Inst. Math.'' , '''90'''  (1969)  ''Trudy Mat. Inst. Steklov.'' , '''90'''  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , Springer  (1982)  (Translated from German)</TD></TR></table>