Geodesic flow

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) of the second kind. A geodesic flow preserves the natural symplectic structure on $TM^n$, while its restriction to $W^{2n-1}$ preserves the corresponding contact structure. Geodesic flows obviously play an important role in geometry (see also 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, to a geodesic flow.

Comments

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

References

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