Gradient dynamical system

A flow (continuous-time dynamical system) given by the gradient of a smooth function $f$ on a smooth manifold. Direct differentiation of $f$ yields a covariant vector (e.g. in the finite-dimensional case in a coordinate neighbourhood $U$ with local coordinates $x^1,\dots,x^n$ this is the vector with components $\partial f/\partial x^1,\dots,\partial f/\partial x^n$), while the phase velocity vector is a contravariant vector. The passage from the one to the other is realized with the aid of a Riemannian metric, and the definition of a gradient dynamical system depends on the choice of the metric (as well as on $f$); the phase velocity vector is often taken with the opposite sign. In the given example the gradient dynamical system in the domain $U$ is described by the system of ordinary differential equations

$$\frac{dx^i}{dt}=\pm\sum_jg^{ij}\frac{\partial f}{\partial x^j},\quad i=1,\dots,n,$$

where the coefficients $g^{ij}$ form a matrix inverse to the matrix of coefficients $\|g_{ij}\|$ of the metric tensor; it is understood that in all $n$ equations the right-hand side is taken with the same "plus" or "minus" sign. A gradient dynamical system is often understood to mean a system of a somewhat more general type [1].

References

[1]	S. Smale, "On gradient dynamical systems" Ann. of Math. (2) , 74 : 1 (1961) pp. 199–206