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On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top. (English) Zbl 1457.70024

Summary: The aim of this work is to show how the moving frames method can be applied for reducing and solving two nonlinear mechanical systems: a bead on a rotating wire hoop and a spinning top. Once both problems are adequately formulated, we explicitly determine the corresponding moving frames associated to the symmetry group of transformations admitted by the systems. The knowledge of the moving frames for the action of the corresponding symmetry groups permits to perform order reductions. Furthermore, we are able to compute the general solutions to each problem from the general solutions of the corresponding reduced systems. Finally, we also discuss the connection of the presented approach with the classical method provided by the celebrated Noether’s Theorem.

MSC:

70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics

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[1] Akbulut, A.; Tascan, F., Application of conservation theorem and modified extended tanh-function method to (1+1)-dimensional nonlinear coupled Klein Gordon Zakharov equation, Chaos, Solitons Fractals, 104, Supplement CSupplement C, 33-40 (2017) · Zbl 1380.35045
[2] Ambrose, DM; Kelliher, JP; Filho, MCL; Lopes, HJN, Serfati solutions to the 2D Euler equations on exterior domains, J. Differ. Equ., 259, 9, 4509-4560 (2015) · Zbl 1321.35144
[3] Anco, S.; Bluman, G., Symmetry and Integration Methods for Differential Equations (2002), New York: Springer, New York · Zbl 1013.34004
[4] Arrigo, DJ, Symmetry Analysis of Differential Equations: An Introduction (2015), Conway: Wiley, Conway · Zbl 1312.00002
[5] Baker, TE; Bill, A., Jacobi elliptic functions and the complete solution to the bead on the hoop problem, Am. J. Phys., 80, 6, 506-514 (2012)
[6] Bartsch, T.; Ding, Y., Periodic solutions of superlinear beam and membrane equations with perturbations from symmetry, Nonlinear Anal. Theory Methods Appl., 44, 6, 727-748 (2001) · Zbl 1088.35004
[7] Basquerotto, C.H.C.C., Righetto, E., da Silva, S.: As simetrias de Lie de um pião. Revista Brasileira de Ensino de Física 40(2) , e2315-1-e2315-8 (2017)
[8] Basquerotto, C.H.C.C., Righetto, E., da Silva, S.: Applications of the Lie symmetries to complete solution of a bead on a rotating wire hoop. J. Braz. Soc. Mech. Sci. Eng. 40(2), 48 (2018)
[9] Basquerotto, CHCC; Ruiz, A.; Righetto, E.; da Silva, S., Moving frames for Lie symmetries reduction of nonholonomic systems, Acta Mech., 230, 8, 2963-2978 (2019) · Zbl 1428.70031
[10] Bocko, J.; Nohajová, V.; Harčarik, T., Symmetries of differential equations describing beams and plates on elastic foundations. Modelling of Mechanical and Mechatronics Systems, Procedia Eng., 48, 40-45 (2012)
[11] Candotti, E.; Palmieri, C.; Vitale, B., On the inversion of Noether’s theorem in classical dynamical systems, Am. J. Phys., 40, 3, 424-429 (1972)
[12] Caruso, F., Estudo da simetria de translação e de suas conseqüê ncias: uma proposta para o ensino médio, Revista Brasileira de Ensino de Física, 30, 3309.1-3309.9 (2008)
[13] Chhay, M.; Hamdouni, A., A new construction for invariant numerical schemes using moving frames, C. R. Méc., 338, 2, 97-101 (2010) · Zbl 1300.76021
[14] Clarkson, PA; Mansfield, EL, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Physica D, 70, 3, 250-288 (1994) · Zbl 0812.35017
[15] Craddock, M., Symmetry groups of linear partial differential equations and representation theory: the Laplace and axially symmetric wave equations, J. Differ. Equ., 166, 1, 107-131 (2000) · Zbl 0962.35010
[16] Estevez, P.; Herranz, F.; de Lucas, J.; Sardón, C., Lie symmetries for Lie systems: applications to systems of odes and pdes, Appl. Math. Comput., 273, 435-452 (2016) · Zbl 1410.34108
[17] Freire, I.L., da Silva, P.L.: Sobre uma classe de equações diferenciais ordinárias admitindo propriedades especiais. In: Anais do Congresso de Matemática Aplicada e Computacional, pp. 622-625 (2013)
[18] Freire, IL; da Silva, PL; Torrisi, M., Lie and Noether symmetries for a class of fourth-order Emden-Fowler equations, J. Phys. A Math. Theor., 46, 24, 245206 (2013) · Zbl 1281.34046
[19] Goldstein, H.; Poole, CP; Safko, J., Classical Mechanics (2011), New York: Pearson, New York
[20] Gonçalves, TMN; Mansfield, EL, Moving frames and conservation laws for euclidean invariant lagrangians, Stud. Appl. Math., 130, 2, 134-166 (2012) · Zbl 1308.37030
[21] Gonçalves, TMN; Mansfield, EL, On moving frames and Noether’s conservation laws, Stud. Appl. Math., 128, 1, 1-29 (2012) · Zbl 1332.37047
[22] Gonçalves, TMN; Mansfield, EL, Moving Frames and Noether’s Conservation Laws—The General Case. Forum of Mathematics, Sigma (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1362.70030
[23] Hydon, P.E.: Symmetry Methods for Differential Equations. Cambridge University Press, (2005) · Zbl 1139.37305
[24] Ibragimov, NH, Practical Course in Differential Equations and Mathematical Modelling, A: Classical and New Methods. Nonlinear Mathematical Models. Symmetry and Invariance Principles (2009), Singapore: World Scientific Publishing Co Pte Ltd, Singapore
[25] Kosmann-Schwarzbach, Y., The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (2011), New York: Springer, New York · Zbl 1216.01011
[26] Lemos, NA, Mecânica analítica (2007), São Paulo: Editora Livraria da Física, São Paulo
[27] Lemos, NA, Analytical Mechanics (2019), Cambridge: Cambridge University Press, Cambridge
[28] Liu, CS, Elastoplastic models and oscillators solved by a Lie-group differential algebraic equations method, Int. J. Non-Linear Mech., 69, 93-108 (2015)
[29] Lutzky, M., Dynamical symmetries and conserved quantities, J. Phys. A Math. Gen., 12, 7, 973 (1979) · Zbl 0413.70005
[30] Mouchet, A., Applications of Noether conservation theorem to Hamiltonian systems, Ann. Phys., 372, Supplement C, 260-282 (2016) · Zbl 1380.70039
[31] Mustafa, M.; Al-Dweik, AY, Noether symmetries and conservation laws of wave equation on static spherically symmetric spacetimes with higher symmetries, Commun. Nonlinear Sci. Numer. Simul., 23, 1-3, 141-152 (2015) · Zbl 1355.58011
[32] Olver, FWJ; Lozier, DW; Boisvert, RF; Clark, CW, NIST Handbook of Mathematical Functions Paperback and CD-ROM (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1198.00002
[33] Olver, PJ, Applications of Lie Groups to Differential Equations (1986), New York: Springer, New York · Zbl 0588.22001
[34] Olver, PJ, Moving frames and singularities of prolonged group actions, Sel. Math. New Ser., 6, 1, 41-77 (2000) · Zbl 0966.57037
[35] Olver, PJ, Generating differential invariants, J. Math. Anal. Appl., 333, 1, 450-471 (2007) · Zbl 1124.53006
[36] Olver, PJ, Equivalence, Invariants and Symmetry (2008), Cambridge: Cambridge University Press, Cambridge
[37] Olver, P.J.: Lectures on moving frames (2012). http://www-users.math.umn.edu/ olver/sm_/mflc.pdf · Zbl 1235.53016
[38] Olver, PJ, Modern developments in the theory and applications of moving frames, Lond. Math. Soc. Impact150 Stories, 1, 14-50 (2015)
[39] Paliathanasis, A.; Tsamparlis, M., Lie point symmetries of a general class of PDEs: the heat equation, J. Geom. Phys., 62, 12, 2443-2456 (2012) · Zbl 1257.35021
[40] Roychowdhury, J., Analyzing circuits with widely separated time scales using numerical PDE methods, IEEE Trans. Circuits Syst. I Fund. Theory Appl., 48, 5, 578-594 (2001) · Zbl 1001.94060
[41] Ruiz, A.; Muriel, C.; Ramírez, J., Exact general solution and first integrals of a remarkable static Euler-Bernoulli beam equation, Commun. Nonlinear Sci. Numer. Simul., 69, 261-269 (2019) · Zbl 1524.34005
[42] Santos, IF, Dinâmica de sistemas mecânicos (2001), São Paul: Ed. Makron Books, São Paul
[43] Sardanashvily, G., Advanced Differential Geometry for Theoreticians (2013), Riga: LAP Lambert Academic Publishing, Riga
[44] Schunck, N.; Dobaczewski, J.; McDonnell, J.; Satua, W.; Sheikh, J.; Staszczak, A.; Stoitsov, M.; Toivanen, P., Solution of the Skyrme Hartree Fock Bogolyubov equations in the cartesian deformed harmonic-oscillator basis, Comput. Phys. Commun., 183, 1, 166-192 (2012)
[45] Soh, CW, Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations, J. Math. Anal. Appl., 345, 1, 387-395 (2008) · Zbl 1146.35004
[46] Stepanova, IV, Symmetry analysis of nonlinear heat and mass transfer equations under Soret effect, Commun. Nonlinear Sci. Numer. Simul., 20, 3, 684-691 (2015) · Zbl 1308.80003
[47] Zhai, XH; Zhang, Y., Noether theorem for non-conservative systems with time delay on time scales, Commun. Nonlinear Sci. Numer. Simul., 52, Supplement C, 32-43 (2017) · Zbl 1510.49019
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