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Definition

A time dependent vector field on a manifold M is a map from an open subset Ω ⊂ R × M {\displaystyle \Omega \subset \mathbb {R} \times M} on T M {\displaystyle TM}

X : Ω ⊂ R × M ⟶ T M ( t , x ) ⟼ X ( t , x ) = X t ( x ) ∈ T x M {\displaystyle {\begin{aligned}X:\Omega \subset \mathbb {R} \times M&\longrightarrow TM\\(t,x)&\longmapsto X(t,x)=X_{t}(x)\in T_{x}M\end{aligned}}}

such that for every ( t , x ) ∈ Ω {\displaystyle (t,x)\in \Omega } , X t ( x ) {\displaystyle X_{t}(x)} is an element of T x M {\displaystyle T_{x}M} .

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

Ω t = { x ∈ M ∣ ( t , x ) ∈ Ω } ⊂ M {\displaystyle \Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M}

is nonempty, X t {\displaystyle X_{t}} is a vector field in the usual sense defined on the open set Ω t ⊂ M {\displaystyle \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:

d x d t = X ( t , x ) {\displaystyle {\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

α : I ⊂ R ⟶ M {\displaystyle \alpha :I\subset \mathbb {R} \longrightarrow M}

such that ∀ t 0 ∈ I {\displaystyle \forall t_{0}\in I} , ( t 0 , α ( t 0 ) ) {\displaystyle (t_{0},\alpha (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 ) ) {\displaystyle {\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 {\displaystyle X} on M {\displaystyle M} can be thought of as a vector field X ~ {\displaystyle {\tilde {X}}} on R × M , {\displaystyle \mathbb {R} \times M,} where X ~ ( t , p ) ∈ T ( t , p ) ( R × M ) {\displaystyle {\tilde {X}}(t,p)\in T_{(t,p)}(\mathbb {R} \times M)} does not depend on t . {\displaystyle t.}

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

R × M ∋ ( t , p ) ↦ ∂ ∂ t | t + X ( p ) ∈ T ( t , p ) ( R × M ) {\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)}

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

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

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

Flow

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

F : D ( X ) ⊂ R × Ω ⟶ M {\displaystyle F:D(X)\subset \mathbb {R} \times \Omega \longrightarrow M}

such that for every ( t 0 , x ) ∈ Ω {\displaystyle (t_{0},x)\in \Omega } ,

t ⟶ F ( t , t 0 , x ) {\displaystyle t\longrightarrow F(t,t_{0},x)}

is the integral curve α {\displaystyle \alpha } of X that satisfies α ( t 0 ) = x {\displaystyle \alpha (t_{0})=x} .

Properties

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

1. If ( t 1 , t 0 , p ) ∈ D ( X ) {\displaystyle (t_{1},t_{0},p)\in D(X)} and ( t 2 , t 1 , F t 1 , t 0 ( p ) ) ∈ D ( X ) {\displaystyle (t_{2},t_{1},F_{t_{1},t_{0}}(p))\in D(X)} then F t 2 , t 1 ∘ F t 1 , t 0 ( p ) = F t 2 , t 0 ( p ) {\displaystyle F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)}
2. ∀ t , s {\displaystyle \forall t,s} , F t , s {\displaystyle F_{t,s}} is a diffeomorphism with inverse F s , t {\displaystyle F_{s,t}} .

Applications

Let X and Y be smooth time dependent vector fields and F {\displaystyle 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 {\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}}

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that η {\displaystyle \eta } 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 {\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}}

This last identity is useful to prove the Darboux theorem.

• Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.