Geodesic flow

In mathematics and physics, the geodetic flow describes a movement along very short connecting routes ( geodesics ). Because geodesics depend not only on their starting point, but also on their starting direction, the geodesic flow can only be defined on the tangential bundle.

definition

Let it be a complete Riemannian manifold with a tangent bundle . According to Hopf-Rinow's theorem, there is a unique geodesic for every tangential vector ${\displaystyle M}$ ${\displaystyle TM}$ ${\displaystyle v\in T_{x}M,x\in M}$

${\displaystyle \gamma _{v}\colon (-\infty ,\infty )\to M}$

With

${\displaystyle \gamma _{v}(0)=x,{\dot {\gamma }}_{v}(0)=v}$.

We define now

${\displaystyle \phi \colon TM\times (-\infty ,\infty )\to TM}$

by

${\displaystyle \phi (v,t):={\dot {\gamma }}_{v}(t)}$.

This defines a flow on , i.e. H. it applies and . ${\displaystyle TM}$ ${\displaystyle \phi (v,s+t)=\phi (\phi (v,t),s)}$ ${\displaystyle \phi (v,0)=v}$

The restriction of the geodetic flow to the unit tangential bundle is often referred to as geodetic flow. ${\displaystyle T^{1}M}$

Geodetic flow as the Hamiltonian river

The geodesic flow is the Hamiltonian flow which in local coordinates passes through

${\displaystyle H(x,p):={\frac {1}{2}}\sum _{i,j}g^{ij}(x)p_{i}p_{j}}$given Hamilton function

${\displaystyle H\colon T^{*}M\to \mathbb {R} }$.

Here the entries denote the matrix inverse to the Riemannian metric . ${\displaystyle g^{ij}(x)}$ ${\displaystyle g_{ij}(x)}$

physics

The Euler's equations for the motion of a rigid body can be interpreted as geodetic flow on the Lie group . ${\displaystyle \mathrm {Isom} (\mathbb {R} ^{3})}$

The Euler equations for fluid dynamics of a inviskosen incompressible flow can be interpreted as geodesic flow on the infinite-dimensional Lie group of maßerhaltenden figures. ${\displaystyle \mathrm {Sdiff} (\mathbb {R} ^{3})}$

Both interpretations go back to Vladimir Arnold .

Geodetic flow on manifolds of non-positive section curvature

In the following, let us be a Riemann manifold of non-positive sectional curvature . ${\displaystyle M}$

Measure theory

The geodesic flow is given the Liouville measure . If is compact, then the geodesic flow is ergodic . Furthermore, in this case it is exponentially mixing , has positive entropy , dense orbits, the set of periodic orbites is dense and there are infinitely many (linearly independent) invariant measures . ${\displaystyle M}$

Stable and unstable manifolds

If has strictly negative section curvature (and even under weaker conditions) the geodetic flow is an Anosov flow . ${\displaystyle M}$

Relationship to the dynamics of the sphere in infinity

If not positive sectional curvature and the geodesic flow has non-migratory , then has the effect of on the sphere at infinity if and dense orbits when the geodesic flow on the Einheitstangentialbündel tight orbits has. ${\displaystyle M}$ ${\displaystyle \pi _{1}M}$ ${\displaystyle \partial _{\infty }{\widetilde {M}}}$

Example: hyperbolic plane

The Poincaré model of the hyperbolic plane , various geodesics ending in the same point (in red) and an associated horocycle (in blue).

Let it be the hyperbolic plane and its unit tangential bundle. The effect of the group of orientation-maintaining isometries ${\displaystyle H^{2}}$ ${\displaystyle T^{1}H^{2}}$

${\displaystyle \mathrm {Isom} ^{+}(H^{2})\simeq {PSL}(2,\mathbb {R} )}$

on induces a bijection between and . We consider the effect of on as a left effect . Then the geodesic flow corresponds to the legal effect of on . The stable and unstable manifolds of the geodetic flow are the restrictions of the unit tangential bundle on the horocycles , algebraically they can be described as the secondary classes of or . ${\displaystyle T^{1}H^{2}}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle T^{1}H^{2}}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle T^{1}H^{2}=PSL(2,\mathbb {R} )}$ ${\displaystyle \Phi _{t}}$ ${\displaystyle \left({\begin{array}{cc}e^{t}&0\\0&e^{-t}\end{array}}\right)}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle N^{+}=\left\{\left({\begin{array}{cc}1&n\\0&1\end{array}}\right):n\in \mathbb {R} \right\}}$ ${\displaystyle N^{-}=\left\{\left({\begin{array}{cc}1&0\\n&1\end{array}}\right):n\in \mathbb {R} \right\}}$

literature

• Patrick Eberlein: Geodesic flows in manifolds of nonpositive curvature. In: Proc. Sympos. Pure Math. Volume 69, 2001, pp. 525-571. (citeseerx.ist.psu.edu; PDF)

Web links

• Terence Tao : The Euler-Arnold equation.
• Terence Tao: Noether's theorem, and the conservation laws for the Euler equations.

Individual evidence

1. ^ V. Arnold: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. In: Ann. Inst. Fourier. (Grenoble). Volume 16, No. 1, 1966, pp. 319-361.
2. ^ DV Anosov : Geodesic flows on closed Riemannian manifolds of negative curvature. In: Trudy Mat. Inst. Steklov. Volume 90, 1967, pp. 3-210. (on-line)
3. ^ W. Ballmann , M. Brin : On the ergodicity of geodesic flows. In: Erg. Th. Dyn. Syst. 2, 1982, pp. 311-315.