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Time dependent vector field
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<h2 id="definition">Definition</h2> <p>A time dependent vector field on a manifold <i>M</i> is a map from an open subset Ω ⊂ R × M {\displaystyle \Omega \subset \mathbb {R} \times M} on T M {\displaystyle TM} </p> 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}}} <p>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} . </p><p>For every t ∈ R {\displaystyle t\in \mathbb {R} } such that the set </p> Ω t = { x ∈ M ∣ ( t , x ) ∈ Ω } ⊂ M {\displaystyle \Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M} <p>is <a href="/facts/Nonempty/ffEk3eA6">nonempty</a>, 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} . </p> <h2 id="associated-differential-equation">Associated differential equation</h2> <p>Given a time dependent vector field <i>X</i> on a manifold <i>M</i>, we can associate to it the following <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a>: </p> d x d t = X ( t , x ) {\displaystyle {\frac {dx}{dt}}=X(t,x)} <p>which is called <a href="/facts/Autonomous_system_(mathematics)/ahlr3A6o">nonautonomous</a> by definition. </p> <h2 id="integral-curve">Integral curve</h2> <p>An <a href="/facts/Integral_curve/wGbVcymt">integral curve</a> of the equation above (also called an integral curve of <i>X</i>) is a map </p> α : I ⊂ R ⟶ M {\displaystyle \alpha :I\subset \mathbb {R} \longrightarrow M} <p>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 <a href="/facts/Domain_of_definition/0vJMw0Nm">domain of definition</a> of <i>X</i> and </p> 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}))} . <h2 id="equivalence-with-time-independent-vector-fields">Equivalence with time-independent vector fields</h2> <p>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.} </p><p>Conversely, associated with a time-dependent vector field X {\displaystyle X} on M {\displaystyle M} is a time-independent one X ~ {\displaystyle {\tilde {X}}} </p> 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)} <p>on R × M . {\displaystyle \mathbb {R} \times M.} In coordinates, </p> X ~ ( t , x ) = ( 1 , X ( t , x ) ) . {\displaystyle {\tilde {X}}(t,x)=(1,X(t,x)).} <p>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. </p> <h2 id="flow">Flow</h2> <p>The <a href="/facts/Flow_(mathematics)/CrmJPfeV">flow</a> of a time dependent vector field <i>X</i>, is the unique differentiable map </p> F : D ( X ) ⊂ R × Ω ⟶ M {\displaystyle F:D(X)\subset \mathbb {R} \times \Omega \longrightarrow M} <p>such that for every ( t 0 , x ) ∈ Ω {\displaystyle (t_{0},x)\in \Omega } , </p> t ⟶ F ( t , t 0 , x ) {\displaystyle t\longrightarrow F(t,t_{0},x)} <p>is the integral curve α {\displaystyle \alpha } of <i>X</i> that satisfies α ( t 0 ) = x {\displaystyle \alpha (t_{0})=x} . </p> <h3>Properties</h3> <p>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)} </p> <ol><li>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)} </li> <li> ∀ t , s {\displaystyle \forall t,s} , F t , s {\displaystyle F_{t,s}} is a <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a> with <a href="/facts/Inverse_function/Tg5nnXlu">inverse</a> F s , t {\displaystyle F_{s,t}} .</li></ol> <h2 id="applications">Applications</h2> <p>Let <i>X</i> and <i>Y</i> be smooth time dependent vector fields and F {\displaystyle F} the flow of <i>X</i>. The following identity can be proved: </p> 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}} <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: </p> 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}} <p>This last identity is useful to prove the <a href="/facts/Darboux_theorem/81zX7bxI">Darboux theorem</a>. </p> <ul><li>Lee, John M., <i>Introduction to Smooth Manifolds</i>, Springer-Verlag, New York (2003) <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-95495-3. Graduate-level textbook on smooth manifolds.</li></ul>