If the vector field represents the force acting on the particle, then by Newton's law you have $F(p(t)) = m \ddot p(t)$, where $m$ is the mass. Since you don't say anything about mass, let's take $m=1$. You can write Newton's equation as a first-order differential equation by setting $u(t)=(p(t),\dot p(t))$ and getting $\dot u = (\dot p, \ddot p) = (\dot p, F(p))$. In this form, Euler's method is simply $u^{k+1} = u^k + h (\dot p^k, F(p^k))$, which gives you two equations: $p^{k+1} = p^k + h \dot p^{k}$, $\dot p^{k+1} = \dot p^{k} + h F(p^k)$, with $p^0 = P$, $\dot p^0 = Q$.