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A flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440901.png" /> whose phase space is the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440902.png" /> of vectors tangent to a Riemannian (more generally, a Finsler) manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440903.png" /> (the so-called configuration manifold of the flow), while the motion is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440904.png" /> be a vector tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440905.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440906.png" /> and let its length be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440907.png" />. Let a geodesic line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440908.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g0440909.png" /> be drawn through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409010.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409011.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409012.png" /> be the point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409013.png" /> the distance of which from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409014.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409015.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409016.png" /> (where that direction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409017.png" /> is considered to be positive which is identical with the direction of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409018.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409019.png" />). One then has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409020.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409021.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409022.png" />. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409023.png" />, and for this reason the vectors of unit length form a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409025.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409026.png" />. A geodesic flow is often understood to mean the restriction of the flow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409027.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409028.png" />. 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
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{{TEX|done}}
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409029.png" /></td> </tr></table>
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$$\frac{d^2x^i}{dt^2}+\sum_{j,k}\Gamma_{jk}^i(x_t)\frac{dx^j}{dt}\frac{dx^k}{dt}=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409030.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409031.png" />-th coordinate of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409032.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409033.png" /> are the Christoffel symbols (cf. [[Christoffel symbol|Christoffel symbol]]) of the second kind. A geodesic flow preserves the natural [[Symplectic structure|symplectic structure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409034.png" />, while its restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409035.png" /> 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.
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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====
 
====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|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044090/g04409036.png" />-system]]). See [[#References|[a1]]]. For applications of geodesic flows in differential geometry, see [[#References|[a3]]], Chapt. 3.
+
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====
 
====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>
 
<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>

Latest revision as of 16:13, 20 September 2014

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)
How to Cite This Entry:
Geodesic flow. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_flow&oldid=18978
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article