Conditional Extremals

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.

Nearly Geodesic Riemannian Cubics in SO(3)

{em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are known for Riemannian cubics, and the asymptotics of Riemannian cubics in SO(3) are reasonably well understood. The mathematical properties and medium-term behaviour of Riemannian cubics in SO(3) are known to be be extremely rich, but there are numerical methods for calculating Riemannian cubic splines in practice. What is missing is an understanding of the short-term behaviour of Riemannian cubics, and it is this that is important for applications. The present paper fills this gap by deriving approximations to nearly geodesic Riemannian cubics in terms of elementary functions. The high quality of these approximations depends on mathematical results that are specific to Riemannian cubics.

Minimum $L^infty$ Accelerations in Riemannian Manifolds

Riemannian cubics are critical points for the $L^2$ norm of acceleration of curves in Riemannian manifolds $M$. In the present paper the $L^infty$ norm replaces the $L^2$ norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are examined when $M$ is a sphere or a bi-invariant Lie group.