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Infinitesimal and infinite numbers in applied mathematics

Aleksandr Bryzgalov, Kevin Islami, Paolo Giordano

TL;DR

Infinitesimal and infinite quantities are widely used informally in applied mathematics; this paper surveys generalized smooth functions (GSF) as a rigorous language for such quantities.It builds the Robinson–Colombeau ring $\widetilde{\mathbb{R}}$, defines GSF with closure under composition, embeds distributions, and proves a Picard–Lindelöf theory for GC$^{k}$ ODEs together with a primitive/integration framework and multidimensional integration in this non-Archimedean setting.It then formalizes deductions that yield heat and wave equations for GSF and demonstrates applications to singular dynamical systems, discontinuous Lagrangians in optics, nonlinear stress–strain models, and quantum-mechanical step and infinite potentials.The results provide a practical, non-Archimedean calculus that reconciles informal infinitesimal reasoning with rigorous mathematics and has broad impact for modeling non-smooth phenomena in physics, engineering, and beyond.

Abstract

The 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.

Infinitesimal and infinite numbers in applied mathematics

TL;DR

Infinitesimal and infinite quantities are widely used informally in applied mathematics; this paper surveys generalized smooth functions (GSF) as a rigorous language for such quantities.It builds the Robinson–Colombeau ring $\widetilde{\mathbb{R}}$, defines GSF with closure under composition, embeds distributions, and proves a Picard–Lindelöf theory for GC$^{k}$ ODEs together with a primitive/integration framework and multidimensional integration in this non-Archimedean setting.It then formalizes deductions that yield heat and wave equations for GSF and demonstrates applications to singular dynamical systems, discontinuous Lagrangians in optics, nonlinear stress–strain models, and quantum-mechanical step and infinite potentials.The results provide a practical, non-Archimedean calculus that reconciles informal infinitesimal reasoning with rigorous mathematics and has broad impact for modeling non-smooth phenomena in physics, engineering, and beyond.

Abstract

The 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.
Paper Structure (26 sections, 39 theorems, 129 equations, 14 figures, 1 table)

This paper contains 26 sections, 39 theorems, 129 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction: informal uses of infinitesimals and infinities in applied mathematics
  2. Numbers: The ring of Robinson-Colombeau
  3. The language of subpoints
  4. Topologies on $\widetilde{\mathbb{R}}^{n}$
  5. Functions and distributions: generalized smooth functions
  6. Definition of GSF and sharp continuity
  7. Embedding of Schwartz distributions and Colombeau generalized functions
  8. Closure with respect to composition
  9. Differential calculus of GSF
  10. Integral calculus using primitives
  11. Classical theorems for GSF
  12. Multidimensional integration
  13. Differential equations: the Picard-Lindelöf theorem for ODE
  14. Formal deductions corresponding to informal reasonings
  15. Derivation of the heat equation for GSF
  16. ...and 11 more sections

Key Result

Lemma 2

Let $x\in\widetilde{\mathbb{R}}$. Then the following are equivalent:

Figures (14)

  • Figure 3.1: A representation of Dirac delta and Heaviside function. A Colombeau mollifier has a representation similar to Dirac delta (but with finite values).
  • Figure 3.2: A representation of $\delta\circ\delta$
  • Figure 6.1: Oscillations of a pendulum wrapping on a parallelepiped
  • Figure 6.2: 8 times re-scaled solution (violet line) in radians and its derivative in rad/s (at $\theta=\theta_{0}=\pi/40$ rad we can see a corner point). Parameters used: $L_{1}=0.4\text{ m}$, $L_{2}=0.2\text{ m}$, $g=9.8\text{ m/$\text{s}^{2}$}$.
  • Figure 6.3: 8 times re-scaled solution in radians (violet line) and its second derivative in rad/$\text{s}^{2}$
  • ...and 9 more figures

Theorems & Definitions (61)

  • Definition 1
  • Lemma 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Definition 8
  • Theorem 9
  • Definition 10
  • ...and 51 more