The Early Period of the Calculus of Variations

Freguglia, Paolo

doi:10.1007/978-3-319-38945-5

Cited by 17 publications

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“…1. Knowing that M(S r ) ≤ + α k (either by (57) or by induction hypothesis) and by (58) we can deduce that S r lies in a ρ k −neighbourhood of S , for ρ k = √ 2rα k .…”

Section: A First Regularity Results For (Almost) Minimizersmentioning

confidence: 90%

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Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

Alouges¹,

Chambolle²,

Stantejsky³

2022

Preprint

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We use the Landau-de Gennes energy to describe a particle immersed into nematic liquid crystals with a constant applied magnetic field. We derive a limit energy in a regime where both line and point defects are present, showing quantitatively that the close-tominimal energy is asymptotically concentrated on lines and surfaces nearby or on the particle. We also discuss regularity of minimizers and optimality conditions for the limit energy.

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“…1. Knowing that M(S r ) ≤ + α k (either by (57) or by induction hypothesis) and by (58) we can deduce that S r lies in a ρ k −neighbourhood of S , for ρ k = √ 2rα k .…”

Section: A First Regularity Results For (Almost) Minimizersmentioning

confidence: 90%

“…The history of interaction between variational problems and geometry has been long and of great mutual influence [28], starting from the geometrically motivated problem of the brachistochrone curve [7,57], Fermat's principle in optics [10], material science [5] to general relativity [30,47].…”

Section: Introductionmentioning

confidence: 99%

Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

Alouges¹,

Chambolle²,

Stantejsky³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Then, among all curves passing through the same pair of endpoints, the desired curve is the one that minimizes the integral M dsv 1/2 . Details on how Euler formulated the principle are provided by Goldstine (1980), and by Freguglia and Giaquinta (2016). 13 Euler remarked: "Since the structure of the universe was made most perfect as designed by the wisest Creator, nothing in the world will occur in which no maximum or minimum rule is shining forth; wherefore there is absolutely no doubt that all the effects of the world can be equally successfully determined from final causes by means of the maximum and least methods, and from the efficient causes themselves."…”

Section: Isoperimetry and Calculus Of Variationsmentioning

confidence: 99%

Dido's Problem. When a myth of ancient literature became a problem of variational calculus

Musielak¹

2023

Preprint

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When introducing the calculus of variations, we may invoke Dido's problem to illustrate the most fundamental variational problem: to find the curve of given perimeter which bounds the greatest area. This type of problem led mathematicians to invent solution methods of maxima and minima, and the genesis of variational calculus as a distinct branch of analysis. Dido's problem was inspired by the mythical tale of the foundation of Carthage (ancient city in North Africa) by a Phoenician princess as told independently by Roman poet Virgil, and by Latin historian Justinus in the first two centuries B.C. Historians have debated the facts surrounding Carthage's birth; however, contemporary mathematicians have accepted the vague events described by Virgil in his Aeneid, adding details to Dido's story to extrapolate a few verses and use as a basis for the isoperimetric theorem. Was Leonhard Euler or Lord Kelvin who first interpreted Virgil's poem as Dido's problem of variational calculus? In this article I attempt to resolve a question of historical attribution to identify who first defined Dido's problem.

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“… Galileo to Cavalieri, 24 February 1640, [Drake (1978. Cavalieri did not take up the problem-"I too left it aside"[Freguglia & Giaquinta (2016), 34]-but Torricelli solved it soon thereafter. 4 Descartes toMersenne, 27 May 1638, AT.II.135, [Jullien (2015, 171].5 [Palmerino & Thijssen (2004), 189].…”

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confidence: 99%

Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution

Blåsjö

2021

Preprint

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I offer a revisionist interpretation of Galileo's role in the history of science. My overarching thesis is that Galileo lacked technical ability in mathematics, and that this can be seen as directly explaining numerous aspects of his life's work. I suggest that it is precisely because he was bad at mathematics that Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His reliance on these hands-on modes of research was not a pioneering contribution to scientific method, but a last resort of a mind ill equipped to make a contribution on mathematical grounds. Likewise, it is precisely because he was bad at mathematics that Galileo expounded at length about basic principles of scientific method. "Those who can't do, teach." The vision of science articulated by Galileo was less original than is commonly assumed. It had long been taken for granted by mathematicians, who, however, did not stop to pontificate about such things in philosophical prose because they were too busy doing advanced scientific work.

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The Early Period of the Calculus of Variations

Cited by 17 publications

References 0 publications

Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field

Dido's Problem. When a myth of ancient literature became a problem of variational calculus

Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution

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