Sur un problème de M. Ulam concernant l'équilibre des corps flottants

Auerbach, H.

doi:10.4064/sm-7-1-121-142

Cited by 47 publications

(80 citation statements)

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“…It has only been proved true for density 0 by Montejano [5], and false for dimension 2 and density 1/2 by Auerbach [1]. In this latter work some remarkable properties about possible examples in dimension 2 were proved.…”

Section: Is a Solid Of Uniform Density Which Will Float In Water In Ementioning

confidence: 84%

“…Next, using the fact that f −1 (a) is an isolated singular point of the vector field and a non-degenerate center, that is, the linear part of the vector field has eigenvalues ±iω, ω > 0, we may prove, using the Classical Poincaré-Lyapunov Center Theorem [3], that the limit of the period function η : (b, a) → R, when A → a, is 2π/ω which, after the corresponding calculations, gives η(a) = 2π/ω ∼ 2.4002. This, together with the fact that a , although we know from [1] that there are with perimetral density …”

Section: An Outline Towards the Classification Of Carousels With Fivementioning

confidence: 99%

“…Let us define, for every t ∈ R, γ(t) = φ(t 1 ), where φ(t 1 ) is the other extreme point of C A (t). It is known (see [1]), that γ is a smooth curve. By Theorem 1, |γ(t) − φ(t)| is constant.…”

Section: ]) a Figure φ(φ) Of Density Floats In Equilibrium In Every mentioning

confidence: 99%

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Carousels, Zindler curves and the floating body problem

2004

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A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.

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Section: Is a Solid Of Uniform Density Which Will Float In Water In Ementioning

confidence: 84%

Section: An Outline Towards the Classification Of Carousels With Fivementioning

confidence: 99%

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Carousels, Zindler curves and the floating body problem

2004

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“…Cylindrical logs of constant two-dimensional cross-section D and specific weight 1/2 do not need to have a disk as their cross section to swim metastable in any orientation parallel to the axis of the cylinder. Any so-called Zindler set (and this includes certain heart-shaped sets D) will serve the same purpose [1]. By definition, Zindler sets have the remarkable property that any line segment dividing the set into two subsets of equal area has the same length, independent of its direction.…”

Section: Open Problemmentioning

confidence: 99%

Two dimensions are easier

Kawohl

2016

Arch. Math.

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“…The problem as formulated is open to some interpretation. Responses containing counterexamples that appeared in [4] and in ensuing literature [5,6,7] were based on hydrostatic theory going back to Archimedes that took no account of surface tension. Independently in 2007 Mattie Sloss and I came on to the problem from a different point of view taking a partial account of surface tension, and published in [3] an affirmative answer to the question in that context, for smooth convex three-dimensional bodies.…”

mentioning

confidence: 99%