Related papers: The hanging chain problem with respect to a circle
The problem of a suspended rope wrapped around a fixed cylinder is studied. If the suspension force is larger than a certain threshold (which is larger than the weight of the rope), the rope would remain tightly wrapped around the cylinder.…
The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved…
A rich zoology of shapes emerges from a simple stretched and twisted elastic ribbon. Despite a lot of interest, all these shape are not understood, in particular the shape that prevails at large tension and twist and that emerges from a…
Leaves and flowers frequently have a characteristic rippling pattern at their edges. Recent experiments found similar patterns in torn plastic. These patterns can be reproduced by imposing metrics upon thin sheets. The goal of this paper is…
A new equation is proposed to explain the curvature of spent sparklers. We found the state of a segment of the sparkler to depend strongly on the state of its spent segments. The equation is nearly able to produce the sparkler shape for a…
We study the motion of a charged particle under the action of a magnetic field with cylindrical symmetry. In particular we consider magnetic fields with constant direction and with magnitude depending on the distance $r$ from the symmetry…
A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
A freely falling chain from a cup at certain height can jump. The process can be divided into two parts: a stable suspension and an accelerating procedure. Variational principle and force analysis demonstrate that the shape of stable…
This paper concerns the number of lattice points in a circle.
The determination of the conductivity of a deterministic or stochastic classical system coupled to reservoirs at its ends can in general be mapped onto the problem of computing the stiffness (the `energy' cost of twisting the boundaries) of…
One cannot pull an open, curved string along itself. This fact is clearly reflected in the unwrapping motion of a string or chain as it is dragged around an object, and implies strong consequences for slender structures in passive…
We propose a new approach for modeling the quantum ring single particle energy spectrum. The approach is based on separation of variables in the Schr\"odinger equation in oblate spheroidal coordinates. We consider a model of a spheroidal…
This paper investigates the hanging chain problem in the simply isotropic plane as well as its 2-dimensional analog in the simply isotropic space. The simply isotropic plane and space are two- and three-dimensional geometries equipped with…
Quantum entanglement is one of the most intriguing phenomena in physics, but many presentations of the subject leave a false impression that it provides a sort of "remote control" for changing the state of a distant particle by local…
We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational…
We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the…
Rectangular cavities are solvable models that nevertheless touch on many of the controversial or mysterious aspects of the vacuum energy of quantum fields. This paper is a thorough study of the two-dimensional scalar field in a rectangle by…
Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures…
The role of geometry on dispersive forces is investigated by calculating the energy between different spheroidal particles and planar surfaces, both with arbitrary dielectric properties. The energy is obtained in the non-retarded limit…
We discuss a pair of isoperimetric problems which at a glance seem to be unrelated. The first one is classical: one places $N$ identical point charges at a closed curve $\Gamma$ at the same arc-length distances and asks about the energy…