zbMATH — the first resource for mathematics

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

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.)
Full Text: DOI Numdam EuDML
[1] Bedford, A., Hamilton’s principle in continuum mechanics, Research Notes in Math., N°139, (1985), Pitman Boston, London, Melbourne · Zbl 0577.73001
[2] Buttazzo, G.; Percivale, D., On the approximation of the elastic bounce problem on Riemannian manifolds, J. Diff. Equations, 47, 227-245, (1983) · Zbl 0498.58011
[3] Fremont, M., Contact with adhesion, (Moreau, J. J.; Panagiotopoulos, P. D.; Strang, G., Topics in Nonsmooth Mechanics, (1988), Birkhäuser Basel)
[4] Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhäuser Boston, Basel, Stuttgart · Zbl 0545.49018
[5] Lichnerowicz, A., Propagateurs et commutateurs en relativité Générale, institut des hautes etudes scientifiques, N° 10, (1961), Publications Mathématiques
[6] Lichnerowicz, A., Théorie des rayons en hydrodynamique et magnétohydrodynamique relativistes, Ann. Inst. Henri Poincaré, 7, 271-302, (1967), Sect. A
[7] Marsden, J. E.; Hughes, T. J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, N.J. · Zbl 0545.73031
[8] Monteiro Marques, M. D.P., Chocs inélastiques standards: un résultat d’existence, travaux du Séminaire d’analyse convexe, vol. 15, (1985), Univ. des Sci. et Techniques du Languedoc Montpellier, exposé n° 4
[9] Monteiro Marques, M. D.P., Inclusões diferenciais e choques inelásticos, doctoral dissertation, faculty of sciences, (1988), University of Lisbon
[10] Moreau, J. J., Fluid dynamics and the calculus of horizontal variations, Int. J. Engng. Sci., 20, 389-411, (1982) · Zbl 0479.76080
[11] Moreau, J. J., Le transport d’une mesure vectorielle par un fluide et le théorème de Kelvin-Helmholtz, Rev. Roum. Math. Pures et Appl., 27, 375-383, (1982) · Zbl 0514.76045
[12] Moreau, J. J., Liaisons unilatérales sans frottement et chocs inélastiques, C.R. Acad. Sci. Paris, 296, 1473-1476, (1983), Sér. II · Zbl 0517.70018
[13] Moreau, J. J., Variational properties of stationary inviscid incompressible flows with possible abrupt inhomogeneity or free surface, Int. J. Engng. Sci., 23, 461-481, (1985) · Zbl 0608.76021
[14] Moreau, J. J., Une formulation de la dynamique classique, C.R. Acad. Sci. Paris, 304, 191-194, (1987), Sér. II · Zbl 0607.70017
[15] Moreau, J. J., Free boundaries and non-smooth solutions to some field equations: variational characterization through the transport method, (Zolesio, J. P., Proceedings of the IFIP WG 8.2 Working Conference on Boundary Control and Boundary Variations, (1988), Springer-Verlag) · Zbl 0664.58003
[16] Moreau, J. J., Bounded variation in time, (Moreau, J. J.; Panagiotopoulos, P. D.; Strang, G., Topics in Nonsmooth Mechanics, (1988), Birkhäuser, Basel)
[17] Moreau, J. J., Unilateral contact and dry friction in finite freedom dynamics, (Moreau, J. J.; Panagiotopoulos, P. D., Non-smooth Mechanics and Applications, (1988), CISM Courses and Lectures, Springer-Verlag Wien) · Zbl 0703.73070
[18] Pandit, S. G.; Deo, S. G., Differential system involving impulses, Lecture Notes in Math., vol. 954, (1982), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0539.34001
[19] Percivale, D., Uniqueness in the elastic bounce problem, J. Diff. Equations, 56, 206-215, (1985) · Zbl 0521.73006
[20] de Rham, G., Variétés différentiables, (1955), Hermann Paris · Zbl 0065.32401
[21] Schatzman, M., A class of nonlinear differential equations of second order in time, J. Nonlinear Analysis, Theory, Methods and Appl., 2, 355-373, (1978) · Zbl 0382.34003
[22] Vol’pert, A. I.; Hudjaev, S. I., Analysis in classes of discontinuous functions and equations of mathematical physics, (1985), Martinus Nijhoff Pub. Dordrecht, Boston, Lancaster · Zbl 0564.46025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.