Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces

• Autorzy: Kovalchuk Vasyl , Gołubowska Barbara , Mladenov Ivaïlo M.

Identyfikatory

DOI

10.24425/bpasts.2021.136727

• Języki publikacji: EN

Abstrakty

EN

In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions to those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic integrals and elliptic functions, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained.

Słowa kluczowe

EN

action-angle analysis; mechanics of infinitesimal gyroscopes; geodesic equations of motion; geodetic equations of motion; helicoid-catenoid deformation family of minimal surfaces; elliptic integrals; elliptic functions

PL

analiza kąta działania; mechanika nieskończenie małych żyroskopów; równanie ruchu geodezyjne; całki eliptyczne; funkcje eliptyczne; rodzina deformacji helikoidów minimalnych powierzchni; rodzina deformacji katenoidów minimalnych powierzchni

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2021
• Tom: Vol. 69, nr 2
• Strony: art. no. e136727
• Opis fizyczny: Bibliogr. 13 poz., rys.

Twórcy

autor

Kovalchuk Vasyl

• Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawińskiego 5B, 02-106 Warsaw, Poland

• https://orcid.org/0000-0001-5391-2706

autor

Gołubowska Barbara

• Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawińskiego 5B, 02-106 Warsaw, Poland

• https://orcid.org/0000-0001-6150-4219

autor

Mladenov Ivaïlo M.

• Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Bl. 21, 1113 Sofia, Bulgaria

Bibliografia

• [1] I.M. Mladenov and M.Ts. Hadzhilazova, “Geometry of the anisotropic minimal surfaces”, An. St. Univ. Ovidius Constanta 20, 79–88 (2012).
• [2] J. Zmrzlikar, Minimal Surfaces in Biological Systems, Faculty of Mathematics and Physics, University of Ljubljana, 2011.
• [3] S.N. Krivoshapko and V.N. Ivanov, Encyclopedia of Analytical Surfaces, Springer, New York-London, 2015.
• [4] A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, Chapman and Hall/CRC, New York, 2006.
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• [6] I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Series, and Products (7th Edition), eds. A. Jeffrey and D. Zwillinger, Academic Press, Oxford, 2007.
• [7] V. Kovalchuk, B. Gołubowska, and I.M. Mladenov, “Mechanics of infinitesimal test bodies on Delaunay surfaces: spheres and cylinders as limits of unduloids and their action-angle analysis”, J. Geom. Symmetry Phys. 53, 55–84, (2019).
• [8] V. Kovalchuk and I.M. Mladenov, “Mechanics of infinitesimal gyroscopes on Mylar balloons and their action-angle analysis”, Math. Meth. Appl. Sci. 43, 3040–3051 (2020).
• [9] J.J. Slawianowski and B. Golubowska, “Bertrand systems on spaces of constant sectional curvature. The action-angle analysis. Classical, quasi-classical and quantum problems”, Geom. Integrability Quantization 16, 110–138 (2015).
• [10] G. De Matteis, L. Martina, C. Naya, and V. Turco, “Helicoids in chiral liquid crystals under external fields”, Phys. Rev. E 100, 05273-(1–12) (2019).
• [11] G. De Matteis, L. Martina, and V. Turco, “Waveguiding by helicoids in confined chiral nematics”, J. Instrum. 15, C05028- (1–11) (2020).
• [12] M. Toda, F. Zhang, and B. Athukorallage, “Elastic surface model for beta-barrels: geometric, computational, and statistical analysis”, Proteins 86, 35–42 (2018).
• [13] J.J. Sławianowski, V. Kovalchuk, B. Gołubowska, A. Martens, and E.E. Rożko, “Dynamical Systems with Internal Degrees of Free- dom in Non-Euclidean Spaces”, IFTR Reports, IPPT PAN, 8/2006.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).