  One can view a partial flag variety in $\mathbb{C}^n$ as an adjoint orbit
$\mathcal{O}_\lambda$ inside the Lie algebra of $n \times n$ skew-Hermitian
matrices. We use the orbit context to study the totally nonnegative part of a
partial flag variety from an algebraic, geometric, and dynamical perspective.
The paper has three main parts:
  (1) We introduce the totally nonnegative part of $\mathcal{O}_\lambda$, and
describe it explicitly in several cases. We define a twist map on it, which
generalizes (in type $A$) a map of Bloch, Flaschka, and Ratiu (1990) on an
isospectral manifold of Jacobi matrices.
  (2) We study gradient flows on $\mathcal{O}_\lambda$ which preserve
positivity, working in three natural Riemannian metrics. In the K\"ahler
metric, positivity is preserved in many cases of interest, extending results of
Galashin, Karp, and Lam (2017, 2019). In the normal metric, positivity is
essentially never preserved on a generic orbit. In the induced metric, whether
positivity is preserved appears to depends on the spacing of the eigenvalues
defining the orbit.
  (3) We present two applications. First, we discuss the topology of totally
nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2017,
2019) showed that the former are homeomorphic to closed balls, and we interpret
their argument in the orbit framework. We also show that a new family of
amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic
to closed balls. Second, we discuss the symmetric Toda flow on
$\mathcal{O}_\lambda$. We show that it preserves positivity, and that on the
totally nonnegative part, it is a gradient flow in the K\"ahler metric up to
applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu
(1990).
