Differentiation of Vector Fields

[Kreizhn]

Homework Statement

Let X,Y be vector fields and x(t) be a curve satisfying
[tex] \dot x(t) = X(x(t)) + u(t) Y(x(t)), u(t) \in \mathbb R [/itex]
and assume there exists p(t) an adjoint curve satisfying
[tex] \dot p(t) = -p(t) \left( \frac{\partial X}{\partial x}(x(t)) + u(t) \frac{\partial Y}{\partial x} (x(t)) \right) [/tex]
If [itex] \langle p(t), Y(x(t)) \rangle = 0 [/itex] show that [itex] \langle p(t), [Y,X](x(t)) \rangle = 0 [/itex]

The Attempt at a Solution

This should be done by differentiating. I get that
[tex] \begin{align*}
\frac{d}{dt} \langle p(t), Y(x(t)) \rangle &= \langle \partial_t p(t), Y(x(t)) \rangle + \langle p(t) \partial_x Y(x(t)) \partial_t x(t) \rangle \\
&= \langle -p \left( \partial_x X +u \partial_x Y\right), Y \rangle + \langle p,\partial_xY ( X + u Y) \rangle \end{align*}
[/tex]

I don't see how this simplifies.

Edit: Where we're taking [itex] [A,B] = (\partial_x A)B - (\partial_x B)A [/itex].

 

[Kreizhn]

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