In mathematics , the flow of a vector field describes the movement along the solution curves of the ordinary differential equation given by the vector field .
definition
Let be a - vector field on an open subset (or more generally on an open subset of a manifold ). According to the existence and uniqueness theorem for ordinary differential equations, there is a unique maximum solution of the differential equation
for each![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![C ^ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91)
![U \ subset \ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1caefb347c86337ea7cd0c354acd2294bd7d81d)
![{\ displaystyle \ gamma _ {x_ {0}} \ colon (a (x_ {0}), b (x_ {0})) \ to \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba15adbcdb08cfc2cb5284378943d1e13c73e77)
-
.
Here is the (possibly infinite) maximum interval on which a solution is defined.
![{\ displaystyle (a (x_ {0}), b (x_ {0}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f8bfa2b5c66a06c0535b3eae7255a088bae234)
Be . Then it's called through
![{\ displaystyle \ Sigma _ {F} = \ left \ {(t, x) \ in \ mathbb {R} \ times U \ colon a (x) <t <b (x) \ right \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89410b1274fef1f1861fbd0b20e23432edcb0839)
![{\ displaystyle \ Phi (t, x): = \ gamma _ {x} (t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/033388e9713e5a637b9bd4720fa8a57febbe8fbd)
given illustration the flow of the vector field .
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
properties
Vector field F (x, y) = (- y, x)
The flow of a vector field is a flow , i. H. a one-parameter transformation group. So it applies
![\ Phi (0, x) = x](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9e28a8cff06417aaade77a86a849dd787747dd)
and
![{\ displaystyle \ Phi (s + t, x) = \ Phi (s, \ Phi (t, x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ed0d00318cea44c498d946b6f317ecf2a3f795)
for everyone .
![x \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906)
example
The flow of the vector field defined on the is given by .
![\ mathbb {R} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd)
![{\ displaystyle F (x, y) = (- y, x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19e32bd50734373c8e5f7c58ff568647796bc06)
![{\ displaystyle \ Phi (t, (x, y)) = (\ cos (t) x + \ sin (t) y, - \ sin (t) x + \ cos (t) y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84ac446a32f4a4973eedb1ba1f4cd38058c6254b)
Complete vector fields
The vector field is called a complete vector field if its flow is defined for all times, i.e. for all , or is equivalent .
![F.](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
![{\ displaystyle a (x_ {0}) = - \ infty, b (x_ {0}) = \ infty}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49a09446365fb584640328a83c303e49db7e215d)
![x_ {0} \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dee8010e46dd5891756fb4b02b43bc661f7c666)
![{\ displaystyle \ Sigma _ {F} = \ mathbb {R} \ times U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a819b0023c99cce6c611b2fd163265c6ed992ffb)
Vector fields with compact carriers are always complete. This is especially true for vector fields on compact manifolds.
literature
Web links