Classical motions of infinitesimal rotators on Mylar balloons

Abstract

This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.

Mechanics of incompressible test bodies moving on λ‐spheres

“…As it was shown in our previous papers, 6–8 instead of quantities , it is convenient to introduce the affine velocities that are corresponding to the internal motion of our infinitesimal test body. They are defined by the relations …”

“…Therefore, we reduce the description of the whole (translational and internal) configuration of the incompressible test body to the one that is based on the pair of time-dependent quantities ( x(t) i , 𝜑(t) A B ) . As it was shown in our previous papers, [6][7][8] instead of quantities 𝜑(t), it is convenient to introduce the affine velocities Ω that are corresponding to the internal motion of our infinitesimal test body. They are defined by the relations…”

λ‐spheres as a new reference model for the geoid

“…For the above generalized Lagrangian function (), we can use the Euler–Lagrange equations in order to find out not only geodesic but also geodetic (i.e., without any potential energy) equations of motion for the test bodies with different internal characteristics (e.g., gyroscopic or incompressible) moving on different two‐dimensional surfaces with nontrivial curvatures (e.g., cylinders and spheres as limiting surfaces for unduloids, 7,8 Mylar balloons, 10 helicoid‐catenoid deformation family of minimal surfaces, 11 and the presently discussed $$$ \lambda $$$‐spheres 4 ).…”

“…We can apply the above‐described mechanical approach in order to find the geodesics for the case of the $$$ \lambda $$$‐spheres. Although we can generally find geodesics directly solving the system of second‐order differential equations () obtained from the Lagrangian function () or () with $$$ I=0 $$$ (see, e.g., Kovalchuk et al 10,11 ), we can also perform the corresponding Legendre transformation to pass from the Lagrangian to Hamiltonian formulation and then obtain the geodesic equations in the form of the first‐order Hamilton's equations (see, e.g., Kovalchuk et al 4,8 ). From the geometrical perspective, these two approaches are equivalent, but from the mechanical one it can be proven that the phase‐space description is more fundamental both in the classical and quantum physics (see, e.g., Sławianowski et al 12 ).…”

Unifying approach to Heisenberg models for spin distributions on curved surfaces of revolution with isothermal (conformal) metrics

“…Next, we can use (as in Kovalchuk and Mladenov 16 ) the definition of the Jacobi's elliptic sine and cosine functions $$$ \mathrm{sn}\left(z,k\right) $$$ and $$$ \mathrm{cn}\left(z,k\right) $$$ as the inverse functions of the incomplete elliptic integral of the first kind $$$ F\left(\theta, k\right) $$$ given by the expression $$$$ F\left(\theta, k\right)=\underset{0}{\overset{\theta }{\int }}\frac{\mathrm{d}\theta }{\sqrt{1-{k}^2{\sin}^2\kern0.2em \theta }}=\underset{0}{\overset{\sin \theta }{\int }}\frac{\mathrm{d}x}{\sqrt{1-{x}^2}\sqrt{1-{k}^2{x}^2}}, $$$$ with $$$ z=\arg \kern3.0235pt \theta $$$ and $$$ \theta =\mathrm{am}\kern3.0235pt z $$$, that is, …”