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Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\gamma \r …

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. …

Think of a beach ball on an pool of water or sand. Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a …

Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation $$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*}\label{ …

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{al …

Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ortho …

New exact solutions in classical general relativity have been found using AdS/CFT methods. It may sound trivial since there are already encyclopedias of exact solutions, but those new ones have quite …

The section 12.2 of Huang's treatise on Statistical Mechanics (1963) gives hints of possible connections between the Ising model and Clifford algebras, but of course $\textit{under the physicist poin …

If some interaction acts differently on the right-handed and left-handed components of a particle, the interaction is automatically parity violating. So you just need to look at the Lagrangian and not …

I have the following PDE in two dimensions $$ 2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0, $$ with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively $$ 2\p …