Analogue to special relativity in other physical systems It's hard to get an intuitive grasp of the Lorentz transformations; I was wondering if the same mathematical formulas – hyperbolic rotation – appear under disguise in other physical systems. Note: I doubt there's a direct mechanical analogy, with a speed limit much lower than $c$ (though that would be great, obviously – perhaps similar in spirit to the optical fibre analogue of event horizons); I'm more thinking of completely different quantities obeying mathematically-equivalent equations. 
 A: Here are two related abstracts that might be of interest to you.
(I have not read the papers to comment.)


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*Geometrical interpretation of optical absorption
J. J. Monzón, A. G. Barriuso, L. L. Sánchez-Soto, and J. M. Montesinos-Amilibia
Phys. Rev. A 84, 023830 – Published 17 August 2011
https://doi.org/10.1103/PhysRevA.84.023830
"We reinterpret the transfer matrix for an absorbing system in very simple geometrical terms. In appropriate variables, the system appears as performing a Lorentz transformation in a (1+3)-dimensional space. Using homogeneous coordinates, we map that action on the unit sphere, which is at the realm of the Klein model of hyperbolic geometry. The effects of absorption appear then as a loxodromic transformation, that is, a rhumb line crossing all the meridians at the same angle."


*Fresnel formulas as Lorentz transformations
Juan José Monzón and Luis L. Sánchez-Soto
Journal of the Optical Society of America A Vol. 17, Issue 8, pp. 1475-1481 (2000) https://doi.org/10.1364/JOSAA.17.001475
"From a matrix formulation of the boundary conditions we obtain the fundamental invariant for an interface and a remarkably simple factorization of the interface matrix, which enables us to express the Fresnel coefficients in a new and compact form. This factorization allows us to recast the action of an interface between transparent media as a hyperbolic rotation. By exploiting the local isomorphism between SL(2, ) and the (3+1)-dimensional restricted Lorentz group SO(3, 1), we construct the equivalent Lorentz transformation that describes any interface."
For different geometric approach to special relativity,
I'll suggest my own approach
"Relativity on Rotated Graph Paper"
American Journal of Physics 84, 344 (2016);
https://doi.org/10.1119/1.4943251
where the emphasis is on a different geometrical figure: the "light-clock diamond", traced out by the spacetime paths of the light signals in a light-clock.
You can play with a visualization at https://www.geogebra.org/m/HYD7hB9v .
A: Yes, there is a direct mechanical analogue in what engineers do with their fingers when they strain a cube of material by parallel-pipeding it.  This is also the same transformation the "cross polarization" of a gravitational wave (GW) does.
Suppose the edges of the cube are the x,y,z axes.  You look down on the xy plane with z axis sticking in your eye. The cube is strained by distorting it into a parallel-piped where the x' axis moves inward to make a small angle $\epsilon_{12}$ radians with the old x axis and the y' axis also moves inward to make a small angle $\epsilon_{12}$ with the old y axis. The quantity $x^2-y^2$ is invariant under the $\epsilon_{12}$ strain. The elements of the matrix which does this strain are the hyperbolic $cosh(\epsilon_{12})$ and $sinh(\epsilon_{12})$.  The strain angle $\epsilon_{12}$ is given by $\frac{dx}{dy}=tanh(\epsilon_{12})$. Successive strains in the xy plane are additive ($\epsilon_{12\ a}+\epsilon_{12\ b}=\epsilon_{12\ total}$).
This mechanical strain in the xy plane (space-space strain) is exactly the same as a Lorentz boost in the xt plane (space-time strain).  The xt axes are strained by distorting it into a parallel-piped where the x' axis moves inward to make a small angle $\lambda_{1}$ radians with the old x axis, and the t' axis also moves inward to make a small angle $\lambda_{1}$ with the old t axis. The quantity $x^2-(ct)^2$ is invariant under the $\lambda_{1}$ strain. The elements of the matrix which does this strain are the hyperbolic $cosh(\lambda_{1})$ and $sinh(\lambda_{1})$. The boost strain angle $\lambda_{1}$ (aka Lorentz Lorentz boost parameter or rapidity) is given by $\frac{dx}{cdt}=tanh(\lambda_{1})$. Successive strains in the xt plane are additive ($\lambda_{1\ a}+\lambda_{1\ b}=\lambda_{1\ total}$). A c is needed to make x and ct have the same dimensions, where as x and y already have the same dimension in the mechanical case, so I guess the "1" in front of y is the analog of the c in front of t.  Both "1" and c are invariant under their respective strains.
If you would like a more complete story (ie: more analogues!) of how rotations and strains make up all the 4x4 matrices (=group GL(4,R)) please see pieces of my old physics stack answers, GL(4,R) and GL(3,R).  Also, this answer GW strains where GWs do the strains rather than the engineer (ie: the $\epsilon$ are just renamed to be the h of GWs).
