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Time dependent vector field

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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.

Contents

Definition

A time dependent vector field on a manifold M is a map from an open subset [\Omega \subset \Bbb \times M] on [TM]

[X: \Omega \subset \Bbb \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] such that the set

[\Omega_t=\ \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=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 \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 \left.}\right|_ =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 \times M] even though its value on a point [(t,x)] does not depend on the component [t \in \Bbb].

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

[\tilde(t,x)=(1,X(t,x))]
for each [(t,x) \in \Omega], where we identify [T_(\Bbb\times M)] with [\Bbb\times T_x M]. We can also write it as:

[ \tilde=\frac}}+X].
To each integral curve of X, we can associate one integral curve of [\tilde], and viceversa.

Flow

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

[F:D(X) \subset \Bbb \times \Omega \longrightarrow M]
such that for every [(t_0,x) \in \Omega],

[t \longrightarrow F(t,t_0,x)]
is the integral curve of X [\alpha] that verifies [\alpha (t_0) = x].

Properties

We define [F_] as [F_(p)=F(t,s,p)]

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

Applications

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

[\frac \left .}\right|_ (F^*_ Y_t)_p = \left( F^*_ \left( [X_,Y_] + \frac \left .}\right|_ 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 \left .}\right|_ (F^*_ \eta_t)_p = \left( F^*_ \left( \mathcal_}\eta_ + \frac \left .}\right|_ \eta_t \right) \right)_p]
This last identity is useful to prove the Darboux theorem.

References

 

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