Functional classical mechanics and rational numbers

  • Anton S. Trushechkin
  • Igor V. Volovich
Research Articles


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.

Key words

classical mechanics Liouville equation rational numbers measurement theory 


  1. 1.
    V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1978).Google Scholar
  2. 2.
    I. V. Volovich, “Number theory as the ultimate physical theory,” preprint, TH 4781/87, CERN, Geneva (1987).Google Scholar
  3. 3.
    I. V. Volovich, “p-Adic string” Class. Quant. Grav. 4, L83–L87 (1987).CrossRefMathSciNetGoogle Scholar
  4. 4.
    E. I. Zelenov, ”Quantum approximation theorem,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 88–90 (2009).CrossRefGoogle Scholar
  5. 5.
    V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).Google Scholar
  6. 6.
    A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer Acad. Publishers, Dordrecht, 1997).MATHGoogle Scholar
  7. 7.
    B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev and I.V. Volovich, “On p-adicmathematical physics,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 1–17 (2009).CrossRefGoogle Scholar
  8. 8.
    V. S. Varadarajan, “Multipliers for the symmetry groups of p-adic spacetime,” p-Adic Numbers, Ultrametric Analysis and Applications 1(1), 69–78 (2009).CrossRefMathSciNetGoogle Scholar
  9. 9.
    I. V. Volovich, “Time irreversibility problem and functional formulation of classical mechanics,” Vestnik Samara State University 8/1(67), 35–55 (2008); arXiv:0907.2445.Google Scholar
  10. 10.
    J. R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements (University Science Books, 1997).Google Scholar
  11. 11.
    B. V. Gnedenko, Theory of Probability (Gordon and Breach Science Publishers, Amsterdam, 1997).MATHGoogle Scholar
  12. 12.
    I. V. Volovich and A. S. Trushechkin, “Squeezed quantum states on an interval and uncertainty relations for nanoscale systems,” Proc. Steklov Inst.Math. 265, 276–306 (2009).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations