Time dependent vector field


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 Bbb{R} imes M on TM

:X: Omega subset Bbb{R} imes 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{}{ ight|_{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} imes 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} imes M, we can associate to it a vector field in the usual sense ilde{X} on Omega such that the autonomous differential equation associated to ilde{X} is essentially equivalent to the nonautonomous differential equation associated to "X". It suffices to impose:

: ilde{X}(t,x)=(1,X(t,x))

for each (t,x) in Omega, where we identify T_{(t,x)}(Bbb{R} imes M) with Bbb{R} imes T_x M. We can also write it as:

: ilde{X}=frac{partial{{partial{t+X.

To each integral curve of "X", we can associate one integral curve of ilde{X}, and viceversa.

Flow

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

:F:D(X) subset Bbb{R} imes 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_{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{}{ ight|_{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{}{ ight|_{t=t_1} Y_t ight) ight)_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{}{ ight|_{t=t_1} (F^*_{t,t_0} eta_t)_p = left( F^*_{t_1,t_0} left( mathcal{L}_{X_{t_1eta_{t_1} + frac{d}{dt} left .{!!frac{}{ ight|_{t=t_1} eta_t ight) ight)_p

This last identity is useful to prove the Darboux theorem.

References

* Lee, John M., "Introduction to Topological Manifolds", Springer-Verlag, New York (2000), ISBN 0-387-98759-2. "Introduction to Smooth Manifolds", Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.


Wikimedia Foundation. 2010.