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

\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}

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

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

\Omega_t=\{x \in M \mid (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)).

Equivalence with time-independent vector fields

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

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

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

on \mathbb{R} \times M. In coordinates,

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

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

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)

  1. 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)
  2. \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.