Uncertainty in space, time, and motion on the special Galilean group

Classical mechanics unfolds within absolute time and Euclidean space, yet our knowledge of where events occur, when they occur, and how motion evolves is inherently uncertain. The special Galilean group provides a natural setting for describing classical spacetime, combining absolute time, Euclidean space, and inertial motion within a single Lie group structure. Although this framework is well known, representing and propagating uncertainty on the group has received comparatively little attention. In this work, we bring together existing results on the structure of the Galilean group and use this unified framework to express uncertainty directly on the group manifold. A main contribution is a compact, closed-form expression for the Galilean group Jacobian, which enables principled uncertainty propagation when composing Galilean transformations. We show that uncertainty in spatial position and orientation, temporal displacement, and inertial motion are intrinsically coupled through the underlying group structure. To illustrate the usefulness of the Galilean framework, we consider the problem of estimating a time-varying transformation between inertial frames from noisy observations collected at distinct instants in time. We show that performing estimation directly on the Galilean group yields substantially more statistically consistent estimates than formulations that treat time independently. Together, these results provide a geometric foundation for reasoning about uncertainty in space, time, and motion in classical mechanics, navigation, and robotics.