Let $\Phi:\mathbb{R}^d\to \mathbb{R}$ be a differentiable.
Suppose we have a curve $\gamma_t$ in $\mathbb{R}^d$, i.e $\gamma : [0,T]\to \mathbb{R}^d$, which evolves according to
\begin{equation} \partial_t \gamma_t = -\nabla \Phi(\gamma_t). \end{equation}
Do we call $\gamma_t$ the flow of $\Phi$? If so why?
This type of problem, when we have a "potential" function $\Phi : \mathbb{R}^d \to \mathbb{R}$, is sometimes called the gradient flow problem. This is because the solution $\gamma$ to the problem flows along the gradient of the potential. It is in contrast to the Newtonian problem when the second derivative is equated to the gradient, i.e. $\partial_t^2 \gamma_t = \pm \nabla \Phi(\gamma_t)$.