Flow of a vector field

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 ${\displaystyle F}$ ${\displaystyle C^{1}}$ ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle x_{0}\in U}$ ${\displaystyle \gamma _{x_{0}}\colon (a(x_{0}),b(x_{0}))\to \mathbb {R} }$

${\displaystyle {\dot {x}}=F(x),x(0)=x_{0}}$.

Here is the (possibly infinite) maximum interval on which a solution is defined. ${\displaystyle (a(x_{0}),b(x_{0}))}$

Be . Then it's called through ${\displaystyle \Sigma _{F}=\left\{(t,x)\in \mathbb {R} \times U\colon a(x)<t<b(x)\right\}}$

${\displaystyle \Phi (t,x):=\gamma _{x}(t)}$

given illustration the flow of the vector field . ${\displaystyle \Phi \colon \Sigma _{F}\to U}$ ${\displaystyle F}$

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

${\displaystyle \Phi (0,x)=x}$

and

${\displaystyle \Phi (s+t,x)=\Phi (s,\Phi (t,x))}$

for everyone . ${\displaystyle x\in U}$

example

The flow of the vector field defined on the is given by . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle F(x,y)=(-y,x)}$ ${\displaystyle \Phi (t,(x,y))=(\cos(t)x+\sin(t)y,-\sin(t)x+\cos(t)y)}$

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 . ${\displaystyle F}$ ${\displaystyle a(x_{0})=-\infty ,b(x_{0})=\infty }$ ${\displaystyle x_{0}\in U}$ ${\displaystyle \Sigma _{F}=\mathbb {R} \times U}$

Vector fields with compact carriers are always complete. This is especially true for vector fields on compact manifolds.

literature

• John Lee: "Introduction to smooth manifolds", Graduate Texts in Mathematics, Springer, ISBN 978-0-387-21752-9
• Vladimir Arnold: "Ordinary differential equations", Universitext, Springer, ISBN 978-3-540-34563-3

Web links

• Flow of a Vectorfield (nLab)