Functional Classical Mechanics and Rational Numbers

TL;DR

This paper proposes a functional formulation of classical mechanics that replaces point trajectories with distribution functions, addressing the physical unobservability of real numbers and providing a more realistic framework.

Contribution

It introduces a new functional approach to classical mechanics based on distribution functions, linking microscopic dynamics to observable rational numbers.

Findings

The fundamental equation is the Liouville equation, not Newton's.

Newton's equation emerges as an approximation for average values.

Constructs probability densities from observable rational measurement results.

Abstract

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A "functional" formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a "beam" of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.