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2. Hjelmslev's Model celebrating Juel's Synthetic Theory of Surfaces.

Author

Lützen, Jesper

Abstract

This note will explain the purpose of a unique mathematical model in the collection of the Department of Mathematics at the University of Copenhagen. It turns out to illustrate a theorem by Christian Juel on the number of lines on synthetic surfaces of order three. [ABSTRACT FROM AUTHOR]

Published
2021
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6. Interactions Between Mathematics and Physics: The History of the Concept of Function-Teaching with and About Nature of Mathematics.

Author

Kjeldsen, Tinne and Lützen, Jesper

Subjects
MATHEMATICS education (Higher), PHYSICS education (Higher), MATHEMATICS teachers, HIGH school teachers, COLLEGE students, HIGHER education
Abstract

In this paper, we discuss the history of the concept of function and emphasize in particular how problems in physics have led to essential changes in its definition and application in mathematical practices. Euler defined a function as an analytic expression, whereas Dirichlet defined it as a variable that depends in an arbitrary manner on another variable. The change was required when mathematicians discovered that analytic expressions were not sufficient to represent physical phenomena such as the vibration of a string (Euler) and heat conduction (Fourier and Dirichlet). The introduction of generalized functions or distributions is shown to stem partly from the development of new theories of physics such as electrical engineering and quantum mechanics that led to the use of improper functions such as the delta function that demanded a proper foundation. We argue that the development of student understanding of mathematics and its nature is enhanced by embedding mathematical concepts and theories, within an explicit-reflective framework, into a rich historical context emphasizing its interaction with other disciplines such as physics. Students recognize and become engaged with meta-discursive rules governing mathematics. Mathematics teachers can thereby teach inquiry in mathematics as it occurs in the sciences, as mathematical practice aimed at obtaining new mathematical knowledge. We illustrate such a historical teaching and learning of mathematics within an explicit and reflective framework by two examples of student-directed, problem-oriented project work following the Roskilde Model, in which the connection to physics is explicit and provides a learning space where the nature of mathematics and mathematical practices are linked to natural science. [ABSTRACT FROM AUTHOR]

Published
2015
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17. Kant, Helmholtz and the Determinacy of Physical Notes.

Author

Cohen, Robert S., Renn, Jürgen, Gavroglu, Kostas, Glick, Thomas F., Grünbaum, Adolf, Schweber, Sylvan S., Stachel, John H., Wartofsky, Marx W., Hendricks, Vincent F., Jørgensen, Klaus Frovin, Lützen, Jesper, Pedersen, Stig Andur, and Hyder, David

Published
2006
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22. Organizing for Total War: DuPont and Smokeless Powder in World War I.

Author

Buchwald, Jed Z., Bos, Henk, Feingold, Mordechai, Franklin, Allan D., Gavroglu, Kostas, Grafton, Anthony, Hoyningen-Huene, Paul, Keller, Evelyn Fox, Levere, Trevor, Lützen, Jesper, Newman, William, Renn, Jürgen, Roland, Alex, Shapiro, Alan, Siraisi, Nancy, Swerdlow, Noel, Macleod, Roy, Johnson, Jeffrey Allan, and Smith, John Kenly

Published
2006
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23. Kuhlmann at War, 1914-1924.

Author

Buchwald, Jed Z., Bos, Henk, Feingold, Mordechai, Franklin, Allan D., Gavroglu, Kostas, Grafton, Anthony, Hoyningen-Huene, Paul, Keller, Evelyn Fox, Levere, Trevor, Lützen, Jesper, Newman, William, Renn, Jürgen, Roland, Alex, Shapiro, Alan, Siraisi, Nancy, Swerdlow, Noel, Macleod, Roy, Johnson, Jeffrey Allan, and Langlinay, Erik

Published
2006
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26. Munitions, the Military, and Chemistry in Russia.

Author

Buchwald, Jed Z., Bos, Henk, Feingold, Mordechai, Franklin, Allan D., Gavroglu, Kostas, Grafton, Anthony, Hoyningen-Huene, Paul, Keller, Evelyn Fox, Levere, Trevor, Lützen, Jesper, Newman, William, Renn, Jürgen, Roland, Alex, Shapiro, Alan, Siraisi, Nancy, Swerdlow, Noel, Macleod, Roy, Johnson, Jeffrey Allan, and Brooks, Nathan M.

Published
2006
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27. Wartime Chemistry in Italy: Industry, the Military, and the Professors.

Author

Buchwald, Jed Z., Bos, Henk, Feingold, Mordechai, Franklin, Allan D., Gavroglu, Kostas, Grafton, Anthony, Hoyningen-Huene, Paul, Keller, Evelyn Fox, Levere, Trevor, Lützen, Jesper, Newman, William, Renn, Jürgen, Roland, Alex, Shapiro, Alan, Siraisi, Nancy, Swerdlow, Noel, Macleod, Roy, Johnson, Jeffrey Allan, and Pancaldi, Giuliano

Published
2006
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29. First World War Explosives Manufacture: The British Experience.

Author

Buchwald, Jed Z., Bos, Henk, Feingold, Mordechai, Franklin, Allan D., Gavroglu, Kostas, Grafton, Anthony, Hoyningen-Huene, Paul, Keller, Evelyn Fox, Levere, Trevor, Lützen, Jesper, Newman, William, Renn, Jürgen, Roland, Alex, Shapiro, Alan, Siraisi, Nancy, Swerdlow, Noel, Macleod, Roy, Johnson, Jeffrey Allan, and Cocroft, Wayne D.

Published
2006
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32. Interactions between mechanics and differential geometry in the 19 century.

Author

Lützen, Jesper

Abstract

79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincaré's geometric and topological reasonings for example about the three body problem [ Poincaré 1890] ( cf. also [ Poincaré 1993], [ Andersson 1994] and [ Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19 century ideas discussed here influenced the developments in the 20 century. Einstein himself is a hotly debated case. Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20 century invention. Klein's view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz. Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions. [ABSTRACT FROM AUTHOR]

Published
1995
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33. Joseph Liouville's work on the figures of equilibrium of a rotating mass of fluid.

Author

Lützen, Jesper

Abstract

After Jacobi's surprising discovery in 1834 that rotating triaxial ellipsoids of fluid could be in equilibrium, Joseph Liouville (1809-1882) began a study of the properties of these figures and of the well known equilibrium ellipsoids of revolution found by Maclaurin. He published six papers on the question, but only a small fraction of his most far-reaching investigations on the stability of the figures of equilibrium, made during the last months of 1842, appeared in print. This paper contains an almost complete reconstruction of Liouville's theory of stability in its historical context. It is based on two manuscripts published here for the first time and on numerous calculations in Liouville's notebooks. Liouville's idea is to determine whether the 'force vive' (the kinetic energy) of a perturbed equilibrium figure has a maximum in the equilibrium state. That he did by expanding the perturbations in a series of Lamé functions. Being largely unpublished, Liouville's theory of stability of equilibrium figures had little impact. Neverthelss the published notes were known to one of his successors, Liapounoff, and the oher successor, Poincaré, was indirectly influenced by Liouville's work on Lamé functions. Liouville had published this work without revealing that it had been developed as a tool for his investigations of the stability of rotating ellipsoids. [ABSTRACT FROM AUTHOR]

Published
1984
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35. Heaviside's operational calculus and the attempts to rigorise it.

Author

Lützen, Jesper

Abstract

At the end of the 19 century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20 century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations ( Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation ( Lévy, Mikusiński). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus. [ABSTRACT FROM AUTHOR]

Published
1979
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