Abstract
AbstractThe need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but tractable models within the famous J. von Neumann reasonably wide area of applicability. We review the theory of generalized smooth functions as a candidate to address both these needs: a rigorous but simple language of infinitesimal and infinite quantities, and the possibility to deal with continuous and generalized function as if they were smooth maps: with pointwise values, free composition and hence nonlinear operations, all the classical theorems of calculus, a good integration theory, and new existence results for differential equations. We exemplify the applications of this theory through several models of singular dynamical systems: deduction of the heat and wave equations extended to generalized functions, a singular variable length pendulum wrapping on a parallelepiped, the oscillation of a pendulum damped by different media, a nonlinear stress–strain model of steel, singular Lagrangians as used in optics, and some examples from quantum mechanics.
Funder
Österreichische Wissenschaftsfonds
Publisher
Springer Science and Business Media LLC
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