  We show that for almost all points on any analytic curve on R^{k} which is
not contained in a proper affine subspace, the Dirichlet's theorem on
simultaneous approximation, as well as its dual result for simultaneous
approximation of linear forms, cannot be improved. The result is obtained by
proving asymptotic equidistribution of evolution of a curve on a strongly
unstable leaf under certain partially hyperbolic flow on the space of
unimodular lattices in R^{k+1}. The proof involves ergodic properties of
unipotent flows on homogeneous spaces.
