Time dependent vector field

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Specialized

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 $\Omega \subset \Bbb{R} \times M$ on $TM$

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

(t,x) \longmapsto X(t,x)=X_t(x) \in T_xM

such that for every $(t,x) \in \Omega$ , $X_t(x)$ is an element of $T_xM$ .

For every $t \in \Bbb{R}$ 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 \Bbb{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 $\Bbb{R} \times M$ even though its value on a point $(t,x)$ does not depend on the component $t \in \Bbb{R}$ .

Conversely, given a time dependent vector field X defined on $\Omega \subset \Bbb{R} \times M$ , 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 $T_{(t,x)}(\Bbb{R}\times M)$ with $\Bbb{R}\times T_x M$ . 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 \Bbb{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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