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On the Gibbs-Liouville theorem in classical mechanics

Henriksson, Andreas (2019) On the Gibbs-Liouville theorem in classical mechanics. [Preprint]

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Abstract

In this article, it is argued that the Gibbs-Liouville theorem is a mathematical representation of the statement that closed classical systems evolve deterministically. From the perspective of an observer of the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that the physical distinctions between the possible states of the system, or, in other words, the information possessed by the observer about the system, is never lost. Thus, it is proposed that the Gibbs-Liouville theorem is a statement about the dynamical evolution of a closed classical system valid in such situations where information about the system is conserved in time. Furthermore, in this article it is shown that the Hamilton equations and the Hamilton principle on phase space follow directly from the differential representation of the Gibbs-Liouville theorem, i.e. that the divergence of the Hamiltonian phase flow velocity vanish. Thus, considering that the Lagrangian and Hamiltonian formulations of classical mechanics are related via the Legendre transformation, it is obtained that these two standard formulations are both logical consequences of the statement of deterministic evolution, or, equivalently, information conservation.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Henriksson, Andreasandreas.henriksson@skole.rogfk.no0000-0001-9014-4320
Keywords: Determinism; Information; Gibbs-Liouville theorem; Hamilton equations; Hamilton principle
Subjects: Specific Sciences > Physics > Classical Physics
Depositing User: Mr. Andreas Henriksson
Date Deposited: 12 May 2019 01:59
Last Modified: 12 May 2019 01:59
Item ID: 16000
Subjects: Specific Sciences > Physics > Classical Physics
Date: 9 May 2019
URI: http://philsci-archive.pitt.edu/id/eprint/16000

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