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''flow''
 
''flow''
  
The action of the additive group of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682101.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682102.png" />.
+
The action of the additive group of real numbers $  \mathbf R $
 +
on a manifold $  M $.
  
Thus, a one-parameter family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682103.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682104.png" /> is a one-parameter transformation group if the following conditions are satisfied:
+
Thus, a one-parameter family $  \{ {\phi _ {t} } : {t \in \mathbf R } \} $
 +
of transformations of a manifold $  M $
 +
is a one-parameter transformation group if the following conditions are satisfied:
  
<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/o/o068/o068210/o0682105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\phi _ {t + s }  x  = \
 +
\phi _ {t} ( \phi _ {s} x),\ \
 +
\phi _ {-} t x  = \
 +
\phi _ {t}  ^ {-} 1 x,\ \
 +
t, s \in \mathbf R ,\ \
 +
x \in M.
 +
$$
  
If the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682106.png" /> is smooth, then the group is usually assumed to be smooth also, that is, the corresponding mapping
+
If the manifold $  M $
 +
is smooth, then the group is usually assumed to be smooth also, that is, the corresponding mapping
  
<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/o/o068/o068210/o0682107.png" /></td> </tr></table>
+
$$
 +
\phi : \mathbf R \times M  \rightarrow  M,\ \
 +
( t, x)  \rightarrow  \phi _ {t} x ,
 +
$$
  
 
is a differentiable mapping of differentiable manifolds.
 
is a differentiable mapping of differentiable manifolds.
  
A more general concept is that of a local one-parameter transformation group of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682108.png" />. It is defined as a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o0682109.png" /> of some open submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821010.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821014.png" />, satisfying the conditions (*) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821016.png" /> for which both sides of the equations are defined.
+
A more general concept is that of a local one-parameter transformation group of a manifold $  M $.  
 +
It is defined as a mapping $  \phi : U \rightarrow M $
 +
of some open submanifold $  U \subset  \mathbf R \times M $
 +
of the form $  U = \cup _ {x \in M }  ( \left ] \epsilon _ {-} ( x), \epsilon _ {+} ( x) \right [ , x) $,  
 +
where $  \epsilon _ {+} ( x) > 0 $,  
 +
$  \epsilon _ {-} ( x) < 0 $
 +
for $  x \in M $,  
 +
satisfying the conditions (*) for all $  t, s \in \mathbf R $,  
 +
$  x \in M $
 +
for which both sides of the equations are defined.
 +
 
 +
With each smooth local one-parameter transformation group  $  \{ \phi _ {t} \} $
 +
of  $  M $
 +
one associates the vector field
  
With each smooth local one-parameter transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821018.png" /> one associates the vector field
+
$$
 +
M  \ni  x  \rightarrow  X _ {x}  = \
 +
\left . {
 +
\frac{d}{dt }
 +
} \phi _ {t} x
 +
\right | _ {t = 0 }  ,
 +
$$
  
<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/o/o068/o068210/o06821019.png" /></td> </tr></table>
+
called the velocity field, or infinitesimal generator, of the group  $  \{ \phi _ {t} \} $.
 +
Conversely, any smooth vector field  $  X $
 +
generates a local one-parameter transformation group  $  \phi _ {t} $
 +
having velocity field  $  X $.  
 +
In local coordinates  $  x  ^ {i} $
 +
on  $  M $
 +
this one-parameter transformation group is given as the solution of the system of ordinary differential equations
  
called the velocity field, or infinitesimal generator, of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821020.png" />. Conversely, any smooth vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821021.png" /> generates a local one-parameter transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821022.png" /> having velocity field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821023.png" />. In local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821025.png" /> this one-parameter transformation group is given as the solution of the system of ordinary differential equations
+
$$
  
<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/o/o068/o068210/o06821026.png" /></td> </tr></table>
+
\frac{d \phi  ^ {i} ( t, x  ^ {j} ) }{dt }
 +
  = X  ^ {i} ( \phi  ^ {j} ( t, x  ^ {k} ) )
 +
$$
  
with the initial conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821028.png" />.
+
with the initial conditions $  \phi  ^ {i} ( 0, x  ^ {j} ) = x  ^ {i} $,  
 +
where $  X = \sum _ {i} X  ^ {i} \partial  / \partial  x  ^ {i} $.
  
If the local one-parameter transformation group generated by the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821029.png" /> can be extended to a global one, then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068210/o06821030.png" /> is called complete. On a compact manifold any vector field is complete, so that there is a one-to-one correspondence between one-parameter transformation groups and vector fields. This is not the case for non-compact manifolds, and the set of complete vector fields is not even closed under addition.
+
If the local one-parameter transformation group generated by the vector field $  X $
 +
can be extended to a global one, then the field $  X $
 +
is called complete. On a compact manifold any vector field is complete, so that there is a one-to-one correspondence between one-parameter transformation groups and vector fields. This is not the case for non-compact manifolds, and the set of complete vector fields is not even closed under addition.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Palais,  "A global formulation of the Lie theory of transformation groups" , Amer. Math. Soc.  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Palais,  "A global formulation of the Lie theory of transformation groups" , Amer. Math. Soc.  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.R. Sell,  "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.R. Sell,  "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold  (1971)</TD></TR></table>

Revision as of 08:04, 6 June 2020


flow

The action of the additive group of real numbers $ \mathbf R $ on a manifold $ M $.

Thus, a one-parameter family $ \{ {\phi _ {t} } : {t \in \mathbf R } \} $ of transformations of a manifold $ M $ is a one-parameter transformation group if the following conditions are satisfied:

$$ \tag{* } \phi _ {t + s } x = \ \phi _ {t} ( \phi _ {s} x),\ \ \phi _ {-} t x = \ \phi _ {t} ^ {-} 1 x,\ \ t, s \in \mathbf R ,\ \ x \in M. $$

If the manifold $ M $ is smooth, then the group is usually assumed to be smooth also, that is, the corresponding mapping

$$ \phi : \mathbf R \times M \rightarrow M,\ \ ( t, x) \rightarrow \phi _ {t} x , $$

is a differentiable mapping of differentiable manifolds.

A more general concept is that of a local one-parameter transformation group of a manifold $ M $. It is defined as a mapping $ \phi : U \rightarrow M $ of some open submanifold $ U \subset \mathbf R \times M $ of the form $ U = \cup _ {x \in M } ( \left ] \epsilon _ {-} ( x), \epsilon _ {+} ( x) \right [ , x) $, where $ \epsilon _ {+} ( x) > 0 $, $ \epsilon _ {-} ( x) < 0 $ for $ x \in M $, satisfying the conditions (*) for all $ t, s \in \mathbf R $, $ x \in M $ for which both sides of the equations are defined.

With each smooth local one-parameter transformation group $ \{ \phi _ {t} \} $ of $ M $ one associates the vector field

$$ M \ni x \rightarrow X _ {x} = \ \left . { \frac{d}{dt } } \phi _ {t} x \right | _ {t = 0 } , $$

called the velocity field, or infinitesimal generator, of the group $ \{ \phi _ {t} \} $. Conversely, any smooth vector field $ X $ generates a local one-parameter transformation group $ \phi _ {t} $ having velocity field $ X $. In local coordinates $ x ^ {i} $ on $ M $ this one-parameter transformation group is given as the solution of the system of ordinary differential equations

$$ \frac{d \phi ^ {i} ( t, x ^ {j} ) }{dt } = X ^ {i} ( \phi ^ {j} ( t, x ^ {k} ) ) $$

with the initial conditions $ \phi ^ {i} ( 0, x ^ {j} ) = x ^ {i} $, where $ X = \sum _ {i} X ^ {i} \partial / \partial x ^ {i} $.

If the local one-parameter transformation group generated by the vector field $ X $ can be extended to a global one, then the field $ X $ is called complete. On a compact manifold any vector field is complete, so that there is a one-to-one correspondence between one-parameter transformation groups and vector fields. This is not the case for non-compact manifolds, and the set of complete vector fields is not even closed under addition.

References

[1] V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)
[2] R. Palais, "A global formulation of the Lie theory of transformation groups" , Amer. Math. Soc. (1957)

Comments

References

[a1] G.R. Sell, "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold (1971)
How to Cite This Entry:
One-parameter transformation group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_transformation_group&oldid=48043
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article