Particle trajectories on manifolds : some remarks on harmonic oscillator on a sphere

• Autorzy: Chyła Kacper S.

Identyfikatory

DOI

10.14708/ma.v47i2.6362

Warianty tytułu

PL

Trajektorie cząstek na rozmaitościach : uwagi o oscylatorze harmonicznym na kuli

• Języki publikacji: EN

Abstrakty

EN

The classical harmonic oscillator with spherical configuration space is studied. The equations of motion are derived from Hamiltonian formulation of mechanics and approximate solutions in some special cases are obtained. A reliable symplectic integrator is used to obtain numerical solutions which are then compared with analytical approximations. The overall impact of positive curvature on the solutions is discussed and illustrated with comparison of solutions on cylindrical and spherical surfaces.

PL

W pracy analizowany jest klasyczny oscylator harmoniczny z sferyczną przestrzenią konfiguracyjną. Równania ruchu pochodzą z formułowania mechaniki Hamiltona i uzyskiwane są przybliżone rozwiązania w niektórych szczególnych przypadkach. Porównano rozwiązania numeryczne z proponowanej metody z przybliżeniami analitycznymi. Omówiono ogólny wpływ dodatniej krzywizny na rozwiazania i zilustrowano przez porównanie rozwiązań na powierzchniach cylindrycznych i sferycznych.

Słowa kluczowe

EN

harmonic oscillations; configuration manifolds; approximate solutions; symplectic integrators

PL

oscylacje harmoniczne; rozmaitości konfiguracji; rozwiązania przybliżone

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Mathematica Applicanda
• Rocznik: 2019
• Tom: Vol. 47, no. 1
• Strony: 45--59
• Opis fizyczny: Bibliogr. 22 poz., fot., rys., wykr.

Twórcy

autor

Chyła Kacper S.

• Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wybrzeże Wyspiańskiego 27, Wrocław 50-370, Poland

Bibliografia

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• [7] B. Z. Chen and W. J. Zhai. Implicit symmetric and symplectic exponentially fitted modified Runge-Kutta-Nyström methods for solving oscillatory problems. J. Inequal. Appl., pages Paper No. 321, 17, 2018. Cited on p. 46.
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• [15] M. F. Rañada and M. Santander. On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane h2. Journal of Mathematical Physics, 43 (1):431-451, 2002. Cited on p. 46.
• [16] M. F. Rañada and M. Santander. On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane h2. II. Journal of Mathematical Physics, 44 (5):2149-2167, 2003. Cited on p. 46.
• [17] J. M. Sanz-Serna. Symplectic integrators for hamiltonian problems: an overview. Acta numerica, 1:243-286, 1992. Cited on p. 46.
• [18] W. Tang, Y. Sun, and J. Zhang. High order symplectic integrators based on continuous-stage Runge-Kutta-Nyström methods. Appl. Math. Comput., 361:670-679, 2019. Cited on p. 46.
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• [21] Y. Wu and B. Wang. Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems. Appl. Math. Lett., 90:215-222, 2019. Cited on p. 46.
• [22] H. Y. Zhai, W. J. Zhai, and B. Z. Chen. A class of implicit symmetric symplectic and exponentially fitted Runge-Kutta-Nyström methods for solving oscillatory problems. Adv. Difference Equ., pages Paper No. 463, 16, 2018. Cited on p. 46.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).