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 [math]\displaystyle{ \Omega \subset \mathbb{R} \times M }[/math] on [math]\displaystyle{ TM }[/math]

[math]\displaystyle{ \begin{align} X: \Omega \subset \mathbb{R} \times M &\longrightarrow TM \\ (t,x) &\longmapsto X(t,x) = X_t(x) \in T_xM \end{align} }[/math]

such that for every [math]\displaystyle{ (t,x) \in \Omega }[/math], [math]\displaystyle{ X_t(x) }[/math] is an element of [math]\displaystyle{ T_xM }[/math].

For every [math]\displaystyle{ t \in \mathbb{R} }[/math] such that the set

[math]\displaystyle{ \Omega_t=\{x \in M \mid (t,x) \in \Omega \} \subset M }[/math]

is nonempty, [math]\displaystyle{ X_t }[/math] is a vector field in the usual sense defined on the open set [math]\displaystyle{ \Omega_t \subset M }[/math].

Associated differential equation

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

[math]\displaystyle{ \frac{dx}{dt}=X(t,x) }[/math]

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

[math]\displaystyle{ \alpha : I \subset \mathbb{R} \longrightarrow M }[/math]

such that [math]\displaystyle{ \forall t_0 \in I }[/math], [math]\displaystyle{ (t_0,\alpha (t_0)) }[/math] is an element of the domain of definition of X and

[math]\displaystyle{ \frac{d \alpha}{dt} \left.{\!\!\frac{}{}}\right|_{t=t_0} =X(t_0,\alpha (t_0)) }[/math].

Equivalence with time-independent vector fields

A time dependent vector field [math]\displaystyle{ X }[/math] on [math]\displaystyle{ M }[/math] can be thought of as a vector field [math]\displaystyle{ \tilde{X} }[/math] on [math]\displaystyle{ \mathbb{R} \times M, }[/math] where [math]\displaystyle{ \tilde{X}(t,p) \in T_{(t,p)}(\mathbb{R} \times M) }[/math] does not depend on [math]\displaystyle{ t. }[/math]

Conversely, associated with a time-dependent vector field [math]\displaystyle{ X }[/math] on [math]\displaystyle{ M }[/math] is a time-independent one [math]\displaystyle{ \tilde{X} }[/math]

[math]\displaystyle{ \mathbb{R} \times M \ni (t,p) \mapsto \dfrac{\partial}{\partial t}\Biggl|_t + X(p) \in T_{(t,p)}(\mathbb{R} \times M) }[/math]

on [math]\displaystyle{ \mathbb{R} \times M. }[/math] In coordinates,

[math]\displaystyle{ \tilde{X}(t,x)=(1,X(t,x)). }[/math]

The system of autonomous differential equations for [math]\displaystyle{ \tilde{X} }[/math] is equivalent to that of non-autonomous ones for [math]\displaystyle{ X, }[/math] and [math]\displaystyle{ x_t \leftrightarrow (t,x_t) }[/math] is a bijection between the sets of integral curves of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \tilde{X}, }[/math] respectively.

Flow

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

[math]\displaystyle{ F:D(X) \subset \mathbb{R} \times \Omega \longrightarrow M }[/math]

such that for every [math]\displaystyle{ (t_0,x) \in \Omega }[/math],

[math]\displaystyle{ t \longrightarrow F(t,t_0,x) }[/math]

is the integral curve [math]\displaystyle{ \alpha }[/math] of X that satisfies [math]\displaystyle{ \alpha (t_0) = x }[/math].

Properties

We define [math]\displaystyle{ F_{t,s} }[/math] as [math]\displaystyle{ F_{t,s}(p)=F(t,s,p) }[/math]

1. If [math]\displaystyle{ (t_1,t_0,p) \in D(X) }[/math] and [math]\displaystyle{ (t_2,t_1,F_{t_1,t_0}(p)) \in D(X) }[/math] then [math]\displaystyle{ F_{t_2,t_1} \circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p) }[/math]
2. [math]\displaystyle{ \forall t,s }[/math], [math]\displaystyle{ F_{t,s} }[/math] is a diffeomorphism with inverse [math]\displaystyle{ F_{s,t} }[/math].

Applications

Let X and Y be smooth time dependent vector fields and [math]\displaystyle{ F }[/math] the flow of X. The following identity can be proved:

[math]\displaystyle{ \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 }[/math]

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that [math]\displaystyle{ \eta }[/math] is a smooth time dependent tensor field:

[math]\displaystyle{ \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 }[/math]

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.