Conditional Extremals
Lyle Noakes

TL;DR
This paper investigates the mathematical properties of conditional extrema, which are optimal interpolations of system trajectories given uncertain prior laws, with explicit solutions in special geometric cases.
Contribution
It provides a foundational analysis of conditional extrema, characterizing their properties under various prior assumptions and deriving explicit solutions in symmetric and Lie group settings.
Findings
Conditional extrema generalize geodesics when prior fields are non-zero.
Explicit solutions involve elliptic functions for symmetric priors.
On Lie groups, conditional extrema relate to 1-parameter subgroups.
Abstract
Imagine that measurements are made at times and of the trajectory of a physical system whose governing laws are given approximately by a class 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 and . An integral curve of an arbitrary vector field is said to be {\em feasible} when it interpolates the measurements. When is critical for distance to , the feasible curve is called a {\em conditional extremum}. When the distance to 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…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Black Holes and Theoretical Physics · Cosmology and Gravitation Theories
