View source for One-parameter transformation group

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''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====
<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====

====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>