Abstract
In this note, we compare the first integrals and exact solutions of equations of motion for scleronomic and rheonomic, and holonomic and nonholonomic oscillators.
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References
L. Bates,, “Problems and Progress in Nonholonomic Reduction,” XXXIII Symposium on Mathematical Physics (Torn, 2001), Rep. Math. Phys., 49, 143–149 (2002).
J. M. Bertrand,, “M\(\grave{\text e}\)moire sur quelques-unes des forms les plus simples que puissent pr\(\grave{\text e}\)senter les int\(\grave{\text e}\)grales des\(\grave{\text e}\)quations diff\(\grave{\text e}\)rentielles du mouvement d\(\acute{}\)un point mat\(\grave{\text e}\)riel,” J. Math. Pures Appl., 2, 113–140 (1857).
A. Bloch,, Nonholonomic Mechanics and Control, Vol. 24, Springer, Interdisciplinary Applied Mathematics, (2015).
A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov, “Nonholonomic Dynamics and Poisson Geometry,” Russian Math. Surveys, 69, 481–538 (2014).
I. A. Bizyaev, A. V. Borisov, and I. S. Mamaev, “Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control,” Regul. Chaot. Dyn., 23, 983–994 (2018).
I. A. Bizyaev, A. V. Borisov, V. V. Kozlov, and I. S. Mamaev, “Fermi-Like Acceleration and Power-Law Energy Growth in Nonholonomic Systems,” Nonlinearity, 32, 3209–3233 (2019).
A. V. Borisov and A. V. Tsiganov, “On Rheonomic Nonholonomic Deformations of the Euler Equations Proposed by Bilimovich,” Preprint arXiv:2002.07670, (2020).
A. V. Borisov, E. A. Mikishanina and A. V. Tsiganov, “On Inhomogeneous Nonholonomic Bilimovich System,” Preprint arXiv:2003.08577, (2020).
O. Bottema, “On the Small Vibrations of Non-Holonomic Systems,” Ind. Math., 11, 296–298 (1949).
M. D. Bustamante and P. Lynch, “Nonholonomic Noetherian Symmetries and Integrals of the Routh Sphere and the Chaplygin Ball,” Regul. Chaot. Dyn., 24, 511–524 (2019).
J. F Cariñena., E. Martínez, and J. Fernández-Núñez, “Noether\(\acute{}\)s Theorem in Time-Dependent Lagrangian Mechanics,” Rep. Math. Phys., 31, 189–203 (1992).
N. G. Chetaev, “On Gauss Principle,” Izv. Fiz.-Mat. Obshch. Kazan Univ., 6, 68–71 (1932-1933).
R. K. Colegrave and M. S. Abdalla,, “Invariants for the Time-Dependent Harmonic Oscillator. I,” J. Phys. A: Math. Gen., 16, 3805–3815 (1983).
R. K. Colegrave, M. S. Abdalla and M. A. Mannan,, “Invariants for the Time-Dependent Harmonic Oscillator. II. Cubic and Quartic Invariants,” J. Phys. A: Math. Gen., 17, 1567–1571 (1984).
G. Darboux, “Sur un probléme de mécanique,” Arch. Néerlandaises Sci., 6, 371–376 (1901).
L. P. Eisenhart, “Separable Systems of Stäckel,” Ann. Math., 35, 284–305 (1934).
V. P. Ermakov, “Second-Order Differential Equations, Conditions of Complete Integrability,” Univ. Izv. Kiev, 20, 1–25 (1880).
J. Grabowski, M. de León, J. C. Marrero, and D. M. de Diego, “Nonholonomic Constraints: a New Viewpoint,” J. Math. Phys., 50, (2009).
Yu. A. Grigoryev and A. V. Tsiganov, “Symbolic Software for Separation of Variables in the Hamilton-Jacobi Equation for the \(L\)-System,” Regul. Chaotic Dyn., 10, 413–422 (2005).
Yu. A. Grigoryev, A. P. Sozonov, and A. V. Tsiganov, “Integrability of Nonholonomic Heisenberg Type Systems,” SIGMA, 12, 112 (2016).
P. Guha,, The Role of the Jacobi Last Multiplier in Nonholonomic Systems and Locally Conformal Symplectic Structure, Diagana T., Toni B. (eds) Mathematical Structures and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham (2018).
F. Fassò and N. Sansonetto, “Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints,” Regul. Chaotic Dyn., 20, 449–462 (2015).
M. de León and D. M. de Diego, “On the Geometry of Non-Holonomic Lagrangian Systems,” J. Math. Phys., 37, 3389–3414 (1996).
M. de León, J. C. Marrero, and D. M. de Diego, “Linear Almost Poisson Structures and Hamilton-Jacobi Equation. Applications to Nonholonomic Mechanics,” J. Geom. Mech., 2, 159–198 (2010).
H. R. Lewis, “Classical and Quantum Systems with Time Dependent Harmonic Oscillator Type Hamiltonians,” Phys. Rev. Lett., 18, 510–512 (1967).
H. R. Lewis and W. B. Riesenfeld, “An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field,” J. Math. Phys., 10, 1458–1473 (1969).
J. R. Ray and J. L. Reid, “More Exact Invariants for the Time-Dependent Harmonic Oscillator,” Phys. Lett. A, 71, 317–318 (1979).
J. L. Reid and J. R. Ray, “Ermakov Systems, Nonlinear Superposition, and Solutions of Nonlinear Equations of Motion,” J. Math. Phys., 21, 1583–1587 (1980).
J. L. Reid and J. R. Ray, “Invariants for Forced Time-Dependent Oscillators and Generalizations,” Phys. Rev. A., 26, 1042–1047 (1982).
P. M. Rios and J. Koiller, Non-Holonomic Systems with Symmetry Allowing a Conformally Symplectic Reduction, New Advances in Celestial Mechanics and Hamiltonian Systems, Springer, US (2004).
R. M. Rosenberg, Analytical Dynamics of Discrete Systems, Plenum Press, New York (1977).
V. V. Rumyantsev,, “The Dynamics of Rheonomic Lagrangian Systems with Constraints,” J. Applied Math. Mech., 48, 380–387 (1984).
Ya. V. Tatarinov, “Consequences of Nonintegrable Perturbation of the Integrable Constraints: Model Problems of Low Dimensionality,” J. Appl. Math. Mech., 51, 579–586 (1987).
B. Vujanovic and S. E. Jones, Variational Methods in Nonconservative Phenomena, Academic Press, Boston (1989).
Funding
This work of A. V. Tsiganov was supported by the Russian Science Foundation (project no. 19-71-30012) and performed at the Steklov Mathematical Institute of the Russian Academy of Sciences. The authors declare that they have no conflicts of interest.
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Tsiganov, A.V. On a Time-Dependent Nonholonomic Oscillator. Russ. J. Math. Phys. 27, 399–409 (2020). https://doi.org/10.1134/S1061920820030115
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DOI: https://doi.org/10.1134/S1061920820030115