Integral curve

In mathematics in the field of differential topology, an integral curve denotes a curve defined on a differentiable manifold , which is closely related to a given smooth vector field on this manifold. For example, electrical field lines represent integral curves of the associated electrical vector field . Ideally, a small Styrofoam ball moves on integral curves of the vector field, which is given by the flow of a river, for example.

definition

Integral curves of a vector field on the two-dimensional unit sphere

Let be a smooth vector field on a manifold of dimension and an arbitrary point. Then a smooth curve on an open interval with an integral curve of through is called if ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle p\in M}$ ${\displaystyle \gamma \colon I\to M}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle t_{0}\in I}$ ${\displaystyle X}$ ${\displaystyle p}$

${\displaystyle \gamma (t_{0})=p;}$

${\displaystyle \gamma '(t)=X(\gamma (t))\quad \forall t\in I.}$

Or in other words: the tangential vector of is identical at every point with the vector given by at this point. ${\displaystyle \gamma }$ ${\displaystyle X}$

existence

In local coordinates the problem is reduced to a system of ordinary differential equations :

${\displaystyle (\gamma ^{i})'(t)=X^{i}(\gamma ^{1}(t),...,\gamma ^{n}(t))}$

where and the smooth functions are on . Together with the boundary condition , it is a classic initial value problem and the Picard-Lindelöf theorem guarantees a unique solution in a neighborhood of . Since solutions of differential equations are often called 'integrals', the term 'integral curve' is appropriate here. ${\displaystyle i=1,...,n}$ ${\displaystyle X^{i}}$ ${\displaystyle M}$ ${\displaystyle \gamma (t_{0})=p}$ ${\displaystyle t_{0}}$

Local flow

For every smooth vector field there is a uniquely determined maximum local flow ${\displaystyle X\colon M\to TM}$

${\displaystyle \varphi \colon A\to M,\quad (t,p)\mapsto \gamma _{p}(t)}$

with the domain of definition

${\displaystyle A=\bigcup \nolimits _{p\in M}I_{p}\times \{p\}\subseteq \mathbb {R} \times M}$.

It is the unique maximal integral curve with and for all . If the manifold is compact , then the flow is global, that is, it applies to all and . ${\displaystyle \gamma _{p}\colon I_{p}\to M}$ ${\displaystyle \gamma _{p}(0)=p}$ ${\displaystyle \gamma _{p}'(t)=X(\gamma _{p}(t))}$ ${\displaystyle t\in I_{p}}$ ${\displaystyle M}$ ${\displaystyle I_{p}=\mathbb {R} }$ ${\displaystyle p\in M}$ ${\displaystyle A=\mathbb {R} \times M}$

literature

• Theodor Bröcker, Klaus Jänich: Introduction to the differential topology . Springer, Berlin 1973, ISBN 3-540-06461-3 , § 8. Dynamic Systems.
• John M. Lee: Introduction to Smooth Manifolds (=  Graduate Texts in Mathematics . No. 218 ). Springer Verlag, New York NY a. a. 2002, ISBN 0-387-95448-1 .

Individual evidence

1. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY a. a. 1988, ISBN 0-387-96790-7 , p. 249.