Functional classical mechanics and rational numbers |
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Authors: | Anton S Trushechkin Igor V Volovich |
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Institution: | (1) Information Sciences, Tokyo University of Science, Yamazaki 2641, 278-8510 Noda, Japan;(2) Mathematical Physics, Steklov Mathematical Institute, Gubkin St 8, 119991 Moscow, Russia; |
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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. |
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