An expression of classical dynamics
type: Item
instance of: scholarly article
title: An expression of classical dynamics
zbMATH Open document ID: 0677.73004
DOI: 10.1016/S0294-1449(17)30022-7
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1989-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78195
review text: The author begins the article by discussing an apparent lack of applications of functional analysis in modern physics. The article consists of redefining concepts of classical continuum mechanics using the Euler variable viewpoint. The author defines tensor measures and introduces the symmetric tensor measure called kinetic tensor measure for a selected subset of space-time called ``a window''. Thus equations of motion become measure differential equations. Thus non-smooth dynamics are generated with a corresponding version of Hamilton's principle. The reviewer points out that alternate formulations have been recently proposed for non-smooth processes. Gel'fand initiated in 1950-s the use of ``rigged Hilbert spaces'' which were used by R. Hoegh-Krohn and Albeverio to generate appropriate measures. Subsequently non-standard techniques were introduced to connect non-smooth and quantized phenomena with classical mechanics [see references in the ``Nonstandard Methods in stochastic analysis and mathematical physics'' by \textit{S. Albeverio}, \textit{J. E. Fenstad}, \textit{R. Høegh-Krohn} and \textit{T. Linstrøm} (1986; Zbl 0605.60005)]. The author offers an important input into recent activity of closing the gap between modern physics and modern mathematics that may be easier to accept by a physicist with some training in measure theory.
Mathematics Subject Classification ID: 74Axx
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 28E05
Mathematics Subject Classification ID: 70H25
Mathematics Subject Classification ID: 46S20
Mathematics Subject Classification ID: 58D20
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 4110215
zbMATH Keywords: Galilean time-space
zbMATH Keywords: four-dimensional vector distribution divergence of kinetic tensor measure
zbMATH Keywords: Riemannian manifold setting of analytical dynamics
zbMATH Keywords: differential operator equilibrium
zbMATH Keywords: transport by test flows
zbMATH Keywords: Euler variable viewpoint
zbMATH Keywords: symmetric tensor measure
zbMATH Keywords: subset of space-time
zbMATH Keywords: window
zbMATH Keywords: measure differential equations
zbMATH Keywords: non-smooth dynamics
reviewed by: Vadim Komkov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4110215