Functional Classical Mechanics and Rational Numbers

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.

8 pages

References in corpus (3)

• p-Adic Mathematical Physics
• Time Irreversibility Problem and Functional Formulation of Classical Mechanics
• Localization properties of squeezed quantum states in nanoscale space domains

Cited by in corpus (8)

• $p$-Adic Mathematical Physics: The First 30 Years
• Hamiltonian of mean force in the weak-coupling and high-temperature approximations and refined quantum master equations
• Randomness in Classical Mechanics and Quantum Mechanics
• Derivation of the particle dynamics from kinetic equations
• Dynamics of wave packets in the functional mechanics
• Semiclassical description of collapses and revivals of quantum wave packets in bounded domains
• Derivation of the Boltzmann equation and entropy production in functional mechanics
• Gaussian approximation and its corrections for driven dissipative Kerr model