  Imagine that measurements are made at times $t_0$ and $t_1$ of the trajectory
of a physical system whose governing laws are given approximately by a class
${\cal A}$ of so-called {\em prior vector fields}. Because the physical laws
are not known precisely, it might be that the measurements are not realised by
the integral curve of any prior field. We want to estimate the behaviour of the
physical system between times $t_0$ and $t_1$.
  An integral curve of an arbitrary vector field $X$ is said to be {\em
feasible} when it interpolates the measurements. When $X$ is critical for $L^2$
distance to ${\cal A}$, the feasible curve is called a {\em conditional
extremum}. When the distance to ${\cal A}$ is actually minimal, the conditional
extremum is a best estimate for the intermediate behaviour of the system.
  The present paper does some of basic groundwork for investigating
mathematical properties of conditional extrema, focusing on cases where ${\cal
A}$ comprises a single prior field $A$. When $A={\bf 0}$ a conditional extremal
is a geodesic arc, but this special case is not very representative. In
general, $A$ enters into the Euler-Lagrange equation for conditional extrema,
and more can be said when $A$ is conservative or has special symmetry.
  We characterise conservative priors on simply-connected Riemannian manifolds
in terms of their conditional extrema: when time is reversed, a constant is
added to the $L^2$ distance. For some symmetric priors on space forms we obtain
conditional extrema in terms of the Weierstrass elliptic function. For
left-invariant priors on bi-invariant Lie groups, conditional extrema are shown
to be right translations of pointwise-products of 1-parameter subgroups.
