Time dependent vector field

In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.

Definition

A time dependent vector field on a manifold M is a map from an open subset $\Omega \subset {\mathbb {R}}\times M$ on $TM$

X:\Omega \subset {\mathbb {R}}\times M\longrightarrow TM

(t,x)\longmapsto X(t,x)=X_{t}(x)\in T_{x}M

such that for every $(t,x)\in \Omega$ , $X_{t}(x)$ is an element of $T_{x}M$ .

For every $t\in {\mathbb {R}}$ such that the set

\Omega _{t}=\{x\in M|(t,x)\in \Omega \}\subset M

is nonempty, $X_{t}$ is a vector field in the usual sense defined on the open set $\Omega _{t}\subset M$ .

Associated differential equation

Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:

{\frac {dx}{dt}}=X(t,x)

which is called nonautonomous by definition.

Integral curve

An integral curve of the equation above (also called an integral curve of X) is a map

\alpha :I\subset {\mathbb {R}}\longrightarrow M

such that $\forall t_{0}\in I$ , $(t_{0},\alpha (t_{0}))$ is an element of the domain of definition of X and

{\frac {d\alpha }{dt}}\left.{\!\!{\frac {}{}}}\right|_{{t=t_{0}}}=X(t_{0},\alpha (t_{0}))

.

Relationship with vector fields in the usual sense

A vector field in the usual sense can be thought of as a time dependent vector field defined on ${\mathbb {R}}\times M$ even though its value on a point $(t,x)$ does not depend on the component $t\in {\mathbb {R}}$ .

Conversely, given a time dependent vector field X defined on $\Omega \subset {\mathbb {R}}\times M$ , we can associate to it a vector field in the usual sense ${\tilde {X}}$ on $\Omega$ such that the autonomous differential equation associated to ${\tilde {X}}$ is essentially equivalent to the nonautonomous differential equation associated to X. It suffices to impose:

{\tilde {X}}(t,x)=(1,X(t,x))

for each $(t,x)\in \Omega$ , where we identify $T_{{(t,x)}}({\mathbb {R}}\times M)$ with ${\mathbb {R}}\times T_{x}M$ . We can also write it as:

{\tilde {X}}={\frac {\partial {}}{\partial {t}}}+X

.

To each integral curve of X, we can associate one integral curve of ${\tilde {X}}$ , and vice versa.

Flow

The flow of a time dependent vector field X, is the unique differentiable map

F:D(X)\subset {\mathbb {R}}\times \Omega \longrightarrow M

such that for every $(t_{0},x)\in \Omega$ ,

t\longrightarrow F(t,t_{0},x)

is the integral curve $\alpha$ of X that satisfies $\alpha (t_{0})=x$ .

Properties

We define $F_{{t,s}}$ as $F_{{t,s}}(p)=F(t,s,p)$

If $(t_{1},t_{0},p)\in D(X)$ and $(t_{2},t_{1},F_{{t_{1},t_{0}}}(p))\in D(X)$ then $F_{{t_{2},t_{1}}}\circ F_{{t_{1},t_{0}}}(p)=F_{{t_{2},t_{0}}}(p)$
$\forall t,s$ , $F_{{t,s}}$ is a diffeomorphism with inverse $F_{{s,t}}$ .

Applications

Let X and Y be smooth time dependent vector fields and $F$ the flow of X. The following identity can be proved:

{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{{t=t_{1}}}(F_{{t,t_{0}}}^{*}Y_{t})_{p}=\left(F_{{t_{1},t_{0}}}^{*}\left([X_{{t_{1}}},Y_{{t_{1}}}]+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{{t=t_{1}}}Y_{t}\right)\right)_{p}

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that $\eta$ is a smooth time dependent tensor field:

{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{{t=t_{1}}}(F_{{t,t_{0}}}^{*}\eta _{t})_{p}=\left(F_{{t_{1},t_{0}}}^{*}\left({\mathcal {L}}_{{X_{{t_{1}}}}}\eta _{{t_{1}}}+{\frac {d}{dt}}\left.{\!\!{\frac {}{}}}\right|_{{t=t_{1}}}\eta _{t}\right)\right)_{p}

This last identity is useful to prove the Darboux theorem.

References

Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.

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