Time dependent vector field - Alchetron, the free social encyclopedia

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 Ω ⊂ R × M on T M

X : Ω ⊂ R × M ⟶ T M

such that for every ( t , x ) ∈ Ω , X t ( x ) is an element of T x M .

For every t ∈ R such that the set

Ω t = { x ∈ M | ( t , x ) ∈ Ω } ⊂ M

is nonempty, X t is a vector field in the usual sense defined on the open set Ω t ⊂ M .

Associated differential equation

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

d x d t = 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

α : I ⊂ R ⟶ M

such that ∀ t 0 ∈ I , ( t 0 , α ( t 0 ) ) is an element of the domain of definition of X and

d α d t | t = t 0 = X ( t 0 , α ( 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 R × M even though its value on a point ( t , x ) does not depend on the component t ∈ R .

Conversely, given a time dependent vector field X defined on Ω ⊂ R × M , we can associate to it a vector field in the usual sense X ~ on Ω such that the autonomous differential equation associated to X ~ is essentially equivalent to the nonautonomous differential equation associated to X. It suffices to impose:

X ~ ( t , x ) = ( 1 , X ( t , x ) )

for each ( t , x ) ∈ Ω , where we identify T ( t , x ) ( R × M ) with R × T x M . We can also write it as:

X ~ = ∂ ∂ t + X .

To each integral curve of X, we can associate one integral curve of X ~ , and vice versa.

Flow

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

F : D ( X ) ⊂ R × Ω ⟶ M

such that for every ( t 0 , x ) ∈ Ω ,

t ⟶ F ( t , t 0 , x )

is the integral curve α of X that satisfies α ( t 0 ) = x .

Properties

We define F t , s as F t , s ( p ) = F ( t , s , p )

If ( t 1 , t 0 , p ) ∈ D ( X ) and ( t 2 , t 1 , F t 1 , t 0 ( p ) ) ∈ D ( X ) then F t 2 , t 1 ∘ F t 1 , t 0 ( p ) = F t 2 , t 0 ( p )
∀ 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:

d d t | t = t 1 ( F t , t 0 ∗ Y t ) p = ( F t 1 , t 0 ∗ ( [ X t 1 , Y t 1 ] + d d t | t = t 1 Y t ) ) p

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that η is a smooth time dependent tensor field:

d d t | t = t 1 ( F t , t 0 ∗ η t ) p = ( F t 1 , t 0 ∗ ( L X t 1 η t 1 + d d t | t = t 1 η t ) ) p

This last identity is useful to prove the Darboux theorem.

References

Time dependent vector field Wikipedia

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