Solutions of a rotated ODE

by Matheus Manzatto   Last Updated August 09, 2018 02:20 AM

I think it's easy but I'm very stuck in this problem

Let $P,Q: \mathbb{R}^2 \to \mathbb{R}$ be smooth funcions and $\varphi(t,(x,y))$ the flow of the differential equation $$\left\{\begin{align*} \dot{x}&=P(x,y)\\ \dot{y}&=Q(x,y). \\ \end{align*}\right.$$ Is it possible to write the flow of $$\left\{\begin{align*} \dot{x}&=Q(x,y)\\ \dot{y}&=-P(x,y) \\ \end{align*}\right.$$ knowing $\varphi(t,(x,y)))$?

Does anyone know how a rotation of $-90^\circ$ in the vector field acts on the solutions of the ODE?

The only thing that I could think of trying (and I failed miserably) was to define $R(x,y) = (y,-x)$ and calculate

\begin{align*}\frac{d}{dt}R(\varphi(t,x)) &= R\left(\frac{d\varphi}{dt} (t,x)\right) = R\left(P(\varphi(t,x),Q(\varphi(t,x)\right) \\&= (Q(\varphi(t,x)),-P(\varphi(t,x)) \neq (Q(R \circ\varphi(t,x)),-P(R\circ\varphi(t,x)) \end{align*}

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