The Problem of Pfaff

Hawkins, Thomas

doi:10.1007/978-1-4614-6333-7_6

Cited by 3 publications

(1 citation statement)

References 39 publications

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“…The second result proven by Stefan and Sussmann in their 1973-1974articles (Sussmann, 1973bStefan, 1974b) is a generalization of Frobenius integrability theorem. This theorem was formulated by the mathematician Frobenius in 1877 as a decisive contribution in the solution to the 'problem of Pfaff' (Hawkins, 2013). This result, in a modern formulation, states that a smooth regular distribution is integrable into a regular foliation if and only if it is involutive 16 .…”

Section: A Community-centered Perspective On Mathematical Knowledgementioning

confidence: 99%

What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

Lavau

2021

trilogía Cienc. Tecnol. Soc.

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This article advocates the benefits of a sociological perspective for the philosophy of mathematical practice. Drawing from the literature of the sociology of sciences, it defends a community-centered approach of the study of mathematical practice and assesses the role of the notion of metamathematics in mathematical change and in stabilized mathematical practices. It relies on the case study of the emergence of geometric control theory at the beginning of the 1970s and of the citational practices associated to the community of control theory since the mid-1990s. The case study shows that the introduction of geometric tools in control theory at the end of the 1960s induced a change in the metamathematical views that control theorists had on their objects. It is then demonstrated how membership to the community of control theory shapes the production and the reception of the theorems of Stefan, Sussmann and Nagano. Interpreting the historical development and citational practices of this community through the perspective of metamathematics, this paper concludes by discussing the role of the orbit theorem in control theory, both as a cognitive label and as a social marker of membership to this community.

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Section: A Community-centered Perspective On Mathematical Knowledgementioning

confidence: 99%

What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

Lavau

2021

trilogía Cienc. Tecnol. Soc.

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A contemporary look at Hermann Hankel’s 1861 pioneering work on Lagrangian fluid dynamics

Frisch

Grimberg²,

Villone

2017

EPJ H

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The present paper is a companion to the paper by Villone and Rampf (2017), titled "Hermann Hankel's On the general theory of motion of fluids, an essay including an English translation of the complete Preisschrift from 1861" together with connected documents. Here we give a critical assessment of Hankel's work, which covers many important aspects of fluid dynamics considered from a Lagrangian-coordinates point of view: variational formulation in the spirit of Hamilton for elastic (barotropic) fluids, transport (we would now say Lie transport) of vorticity, the Lagrangian significance of Clebsch variables, etc. Hankel's work is also put in the perspective of previous and future work. Hence, the action spans about two centuries: from Lagrange's 1760-1761 Turin paper on variational approaches to mechanics and fluid mechanics problems to Arnold's 1966 founding paper on the geometrical/variational formulation of incompressible flow. The 22-year old Hankel -who was to die 12 years later -emerges as a highly innovative master of mathematical fluid dynamics, fully deserving Riemann's assessment that his Preisschrift contains "all manner of good things."

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Comment on Clebsch’s 1857 and 1859 papers on using Hamiltonian methods in hydrodynamics

Grimberg¹,

Tassi

2021

EPJ H

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The present paper is a companion of two translated articles by Alfred Clebsch, titled "On a general transformation of the hydrodynamical equations" and "On the integration of the hydrodynamical equations". The originals were published in the Journal für die reine and angewandte Mathematik" (1857 and 1859). Here we provide a detailed critical reading of these articles, which analyzes methods, and results of Clebsch. In the first place, we try to elucidate the algebraic calculus used by Clebsch in several parts of the two articles that we believe to be the most significant ones. We also provide some proofs that Clebsch did not find necessary to explain, in particular concerning the variational principles stated in his two articles and the use of the method of Jacobi's Last Multiplier. When possible, we reformulate the original expressions by Clebsch in the language of vector analysis, which should be more familiar to the reader. The connections of the results and methods by Clebsch with his scientific context, in particular with the works of Carl Jacobi, are briefly discussed. We emphasize how the representations of the velocity vector field conceived by Clebsch in his two articles, allow for a variational formulation of hydrodynamics equations in the steady and unsteady case. In particular, we stress that what is nowadays known as the "Clebsch variables", permit to give a canonical Hamiltonian formulation of the equations of fluid mechanics. We also list a number of further developments of the theory initiated by Clebsch, which had an impact on presently active areas of research, within such fields as hydrodynamics and plasma physics.

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The Problem of Pfaff

Cited by 3 publications

References 39 publications

What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

What Is a Theorem (in Practice)? The Role of Metamathematics in the Making of Mathematics

A contemporary look at Hermann Hankel’s 1861 pioneering work on Lagrangian fluid dynamics

Comment on Clebsch’s 1857 and 1859 papers on using Hamiltonian methods in hydrodynamics

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