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Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light…

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain D are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving…

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The…

For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description…

We study various formulations of the boundary conditions for the Euler equations of gas dynamics from a mathematical and numerical point of view. In the case of one space dimension, we recall the classical results, based on an analysis of…

In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the…

In this paper, we study the inverse surfaces in 3-dimensional Euclidean space $\mathbb{E}^{3}$. We obtain some results relating Christoffel symbols, the normal curvatures, the shape operators and the third fundamental forms of the inverse…

The energy of extended classical objects, such as vortices, depends on their shape. In particular, we show that the curvature energy of a kink in two spatial dimensions, as a prototype of extended classical solutions, is always negative. We…

A stationary rotating surface is a compact surface in Euclidean space whose mean curvature $H$ at each point $x$ satisfies $2H(x)=a r^2+b$, where $r$ is the distance from $x$ to a fixed straight-line $L$, and $a$ and $b$ are constants.…

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the…

An instantaneous sub-surface disturbance in a two-dimensional elastic half-space is considered. The disturbance propagates through the elastic material until it reaches the free surface, after which it propagates out along the surface. In…

We consider the dynamics of moving curves in three-dimensional Minkowski space $R_1^{3}$ and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space…

We introduce a local coordinate description for the correspondence between the space of oriented affine lines in Euclidean ${\Bbb{R}}^3$ and the tangent bundle to the 2-sphere. These can be utilised to give canonical coordinates on surfaces…

We consider the problem of minimizing Euler's elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values +1 on the inside and -1 on the outside of…

We study the evolution of a self-gravitating compressible fluid in spherical symmetry and we prove the existence of weak solutions with bounded variation for the Einstein-Euler equations of general relativity. We formulate the initial value…

We derive an explicit formula for global weak solutions of the one dimensional system of pressure-less Euler-Poisson equations. Our variational formulation is an extension of the well-known formula for entropy solutions of the scalar…

Using the properties of the angular momentum, we develop a new geometrical technique to study relative equilibria for a system of $3$--bodies with positive masses, moving on the two sphere under the influence of an attractive potential…

We show that there exist closed three-dimensional Riemannian manifolds where the incompressible Euler equations exhibit smooth steady solutions that are isolated in the $C^1$-topology. The proof of this fact combines ideas from dynamical…

We analyse the scaling properties of the energy spectra in fully developed incompressible turbulence in forced, rotating fluids in three dimensions (3D), which are believed to be characterised by universal scaling exponents in the inertial…

We study a mathematical model for deformation of glued elastic bodies in 2D or 3D, which is a linear elasticity system with adhesive force on the glued surface. We reveal a variational structure of the model and prove the unique existence…