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An expression of classical dynamics. (English) Zbl 0677.73004
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 S. Albeverio, J. E. Fenstad, R. Høegh-Krohn and 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.
Reviewer: V.Komkov

MSC:
74Axx Generalities, axiomatics, foundations of continuum mechanics of solids
49J52 Nonsmooth analysis
28E05 Nonstandard measure theory
70H25 Hamilton’s principle
46S20 Nonstandard functional analysis
58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
58A30 Vector distributions (subbundles of the tangent bundles)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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