Orbital derivative
= Orbital derivative
{title2=$\dot V$}
The <orbital derivative> of a differentiable scalar $V$ along an <autonomous differential equation> $\dot x=f(x)$ is
$$
\dot V(x)=\nabla V(x)\cdot f(x).
$$
The <chain rule> identifies it with $dV(x(t))/dt$. It measures change along a trajectory, rather than the size of the gradient alone.
For $V(x,y)=(x^2+y^2)/2$ and the rotation field $f=(-y,x)$, it is $-xy+yx=0$ even though the gradient is nonzero away from the origin. For the contracting field $f=(-x,-y)$, the same function has orbital derivative $-x^2-y^2$. This directional distinction is what makes an orbital derivative useful in a <Lyapunov function> or a <first integral>.