Deterministic physical systems under uncertain initial conditions: the case of maximum entropy applied to projectile motion

TL;DR

This paper applies the Maximum Entropy inference method to projectile motion, demonstrating how to infer initial conditions from average range data, thereby reducing uncertainty and gaining insights into the system's probabilistic behavior.

Contribution

It introduces a MaxEnt-based approach to infer initial conditions of deterministic systems with uncertain initial states, providing a new tool for physics analysis.

Findings

MaxEnt effectively infers initial conditions from average range data.

The method reduces uncertainty in initial conditions.

It offers insights into the probability distribution of projectile paths.

Abstract

The kinematics and dynamics of deterministic physical systems have been a foundation of our understanding of the world since Galileo and Newton. For real systems, however, uncertainty is largely present via external forces such as friction or lack of precise knowledge about the initial conditions of the system. In this work we focus in the latter case and describe the use of inference methodologies in solving the statistical properties of classical systems subject to uncertain initial conditions. In particular we describe the application of the formalism of Maximum Entropy (MaxEnt) inference to the problem of projectile motion given information about the average horizontal range over many realizations. By using MaxEnt we can invert the problem and use the provided information on the average range to reduce the original uncertainty in the initial conditions, while also achieving additional insights based on the shape of the posterior probabilities for the initial conditions probabilities and the projectile path distribution itself. The wide applicability of this procedure, as well as its ease of use, reveals a useful tool by which to revisit a large number of physics problems, from classrooms to frontier research.