Differentiation of Vector Fields

[Kreizhn]

Homework Statement

Let X,Y be vector fields and x(t) be a curve satisfying
\dot x(t) = X(x(t)) + u(t) Y(x(t)), u(t) \in \mathbb R [/itex]<br /> and assume there exists p(t) an adjoint curve satisfying<br /> \dot p(t) = -p(t) \left( \frac{\partial X}{\partial x}(x(t)) + u(t) \frac{\partial Y}{\partial x} (x(t)) \right)<br /> If \langle p(t), Y(x(t)) \rangle = 0 show that \langle p(t), [Y,X](x(t)) \rangle = 0<br /> <br /> <h2>The Attempt at a Solution</h2><br /> This should be done by differentiating. I get that<br /> \begin{align*}&lt;br /&gt; \frac{d}{dt} \langle p(t), Y(x(t)) \rangle &amp;amp;= \langle \partial_t p(t), Y(x(t)) \rangle + \langle p(t) \partial_x Y(x(t)) \partial_t x(t) \rangle \\&lt;br /&gt; &amp;amp;= \langle -p \left( \partial_x X +u \partial_x Y\right), Y \rangle + \langle p,\partial_xY ( X + u Y) \rangle \end{align*}&lt;br /&gt;<br /> <br /> I don&#039;t see how this simplifies.<br /> <br /> Edit: Where we&#039;re taking [A,B] = (\partial_x A)B - (\partial_x B)A.

 

[Kreizhn]

It works if the vector fields are somehow self-adjoint, but I have no reason to believe that this should be true.