Abstract
We present an algorithm for constructing analytically approximate integrals of motion in simple time-periodic Hamiltonians of the form \(H=H_{0}+\varepsilon H_{i}\), where \(\varepsilon\) is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different values of \(\varepsilon\). We find the values of \(\varepsilon_{crit}\) beyond which the orbits escape to infinity and construct integrals which are expressed as series in the perturbation parameter \(\varepsilon\) and converge up to \(\varepsilon_{crit}\). In the absence of resonances the invariant curves are concentric ellipses which are approximated very well by our integrals. Finally, we construct an integral of motion which describes the hyperbolic stroboscopic invariant curve of a resonant case.
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References
Contopoulos, G. and Moutsoulas, M., Resonance Cases and Small Divisors in a Third Integral of Motion: 2, Astron. J., 1965, vol. 70, no. 10, pp. 817–835.
Gustavson, F. G., On Constructing Formal Integrals of a Hamiltonian System near an Equilibrium Point, Astron. J., 1966, vol. 71, no. 8, pp. 670–686.
Giorgilli, A., A Computer Program for Integrals of Motion, Comput. Phys. Commun., 1979, vol. 16, no. 3, pp. 331–343.
Efthymiopoulos, Ch. and Sándor, Zs., Optimized Nekhoroshev Stability Estimates for the Trojan Asteroids with a Symplectic Mapping Model of Co-Orbital Motion, Mon. Not. R. Astron. Soc., 2005, vol. 364, no. 1, pp. 253–271.
Contopoulos, G., Adiabatic Invariants and the “Third” Integral, J. Mathematical Phys., 1966, vol. 7, pp. 788–797.
Markeyev, A. P., Third-Order Resonance in a Hamiltonian System with One Degree of Freedom, J. Appl. Math. Mech., 1994, vol. 58, no. 5, pp. 793–804; see also: Prikl. Mat. Mekh., 1994, vol. 58, no. 5, pp. 37-48.
Markeev, A. P., Stability of Equilibrium States of Hamiltonian Systems: A Method of Investigation, Mech. Solids, 2004, vol. 39, no. 6, pp. 1–8.
Markeev, A. P., On a Multiple Resonance in Linear Hamiltonian Systems, Dokl. Phys., 2005, vol. 50, no. 5, pp. 278–282; see also: Dokl. Akad. Nauk, 2005, vol. 402, no. 3, pp. 339-343.
Markeyev, A. P., Multiple Parametric Resonance in Hamiltonian Systems, J. Appl. Math. Mech., 2006, vol. 70, no. 2, pp. 176–194; see also: Prikl. Mat. Mekh., 2006, vol. 70, no. 2, pp. 200-220.
Markeev, A. P., On the Birkhoff Transformation in the Case of Complete Degeneracy of Quadratic Part of the Hamiltonian, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 309–316.
Kholostova, O. V., Non-Linear Oscillations of a Hamiltonian System with One Degree of Freedom and Fourth-Order Resonance, J. Appl. Math. Mech., 1998, vol. 62, no. 6, pp. 883–892; see also: Prikl. Mat. Mekh., 1998, vol. 62, no. 6, pp. 957-967.
Kholostova, O. V., The Periodic Motions of a Non-Autonomous Hamiltonian System with Two Degrees of Freedom at Parametric Resonance of the Fundamental Type, J. Appl. Math. Mech., 2002, vol. 66, no. 4, pp. 529–538; see also: Prikl. Mat. Mekh., 2002, vol. 66, no. 4, pp. 540-551.
Kholostova, O. V., Resonant Periodic Motions of Hamiltonian Systems with One Degree of Freedom When the Hamiltonian Is Degenerate, J. Appl. Math. Mech., 2006, vol. 70, no. 4, pp. 516–526; see also: Prikl. Mat. Mekh., 2006, vol. 70, no. 4, pp. 568-580.
Bardin, B. and Lanchares, V., On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 627–648.
Bruno, A. D., Normal Form of a Hamiltonian System with a Periodic Perturbation, Comput. Math. Math. Phys., 2020, vol. 60, no. 1, pp. 36–52.
Bruno, A. D., Normalization of a Periodic Hamiltonian System, Program. Comput. Softw., 2020, vol. 46, no. 2, pp. 76–83.
Kandrup, H. E. and Drury, J., Chaos in Cosmological Hamiltonians, Ann. N. Y. Acad. Sci., 1998, vol. 867, no. 1, pp. 306–320.
Kandrup, H. E., Vass, I. M., and Sideris, I. V., Transient Chaos and Resonant Phase Mixing in Violent Relaxation, Mon. Not. R. Astron. Soc., 2003, vol. 341, no. 3, pp. 927–936.
Terzić, B. and Kandrup, H. E., Orbital Structure in Oscillating Galactic Potentials, Mon. Not. R. Astron. Soc., 2004, vol. 347, no. 3, pp. 957–967.
Efthymiopoulos, C. and Contopoulos, G., Chaos in Bohmian Quantum Mechanics, J. Phys. A, 2006, vol. 39, no. 8, pp. 1819–1852.
McLachlan, N. W., Theory and Application of Mathieu Functions, Oxford: Clarendon, 1951.
Richards, J. A., Analysis of Periodically Time-Varying Systems, Berlin: Springer, 1983.
Mathieu, É., Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique, J. Math. Pure Appl., 1868, vol. 13, pp. 137–203.
Leibscher, M. and Schmidt, B., Quantum Dynamics of a Plane Pendulum, Phys. Rev. A, 2009, vol. 80, no. 1, 012510, 16 pp.
Birkandan, T. and Hortaçsu, M., Examples of Heun and Mathieu Functions As Solutions of Wave Equations in Curved Spaces, J. Phys. A, 2007, vol. 40, no. 5, pp. 1105–1116.
Fink, J. K., Physical Chemistry in Depth, Berlin: Springer, 2009.
Ruby, L., Applications of the Mathieu Equation, Am. J. Phys., 1996, vol. 64, no. 1, pp. 39–44.
Contopoulos, G., Resonance Cases and Small Divisors in a Third Integral of Motion: 1, Astronom. J., 1963, vol. 68, pp. 763–779.
Contopoulos, G., Order and Chaos in Dynamical Astronomy, Berlin: Springer, 2002.
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MSC2010
70H05, 70H12
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Tzemos, A.C., Contopoulos, G. Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation. Regul. Chaot. Dyn. 26, 89–104 (2021). https://doi.org/10.1134/S1560354721010056
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DOI: https://doi.org/10.1134/S1560354721010056