  Principal curves are natural generalizations of principal lines arising as
first principal components in the Principal Component Analysis. They can be
characterized from a stochastic point of view as so-called self-consistent
curves based on the conditional expectation and from the variational-calculus
point of view as saddle points of the expected difference of a random variable
and its projection onto some curve, where the current curve acts as argument of
the energy functional. Beyond that, Duchamp and St\"utzle (1993,1996) showed
that planar curves can by computed as solutions of a system of ordinary
differential equations. The aim of this paper is to generalize this
characterization of principal curves to $\mathbb R^d$ with $d \ge 3$. Having
derived such a dynamical system, we provide several examples for principal
curves related to uniform distribution on certain domains in $\mathbb R^3$.
