Adiabatic invariants for strongly nonlinear dynamical systems described with complex functions
Author:
L. Cveticanin
Journal:
Quart. Appl. Math. 54 (1996), 407-421
MSC:
Primary 34C29; Secondary 34C99, 70H05, 70K99
DOI:
https://doi.org/10.1090/qam/1402402
MathSciNet review:
MR1402402
Full-text PDF Free Access
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Abstract: In this paper the adiabatic invariants for strongly nonlinear dynamical systems with two degrees of freedom described by complex functions are obtained. The method [1] developed for dynamical systems with one degree of freedom is extended to systems with two degrees of freedom. The method is based on Noether’s theory and the use of Krylov-Bogolubov-Mitropolski (KBM) and elliptic-Krylov-Bogolubov (EKB) asymptotic techniques. The adiabatic invariants for two types of strong nonlinearities are constructed: the pure cubic nonlinearity and quasi-cubic nonlinearity. The adiabatic invariants are used to obtain the approximate solution to the equations of motion.
Dj. S. Djukic, Adiabatic invariants for dynamical systems with one degree of freedom, Internat. J. Non-Linear Mech. 16, 489–498 (1981)
P. G. L. Leach, Invariants and wave functions for some time-dependent harmonic oscillator-type Hamiltonians, J. Math. Phys. 18, 1902–1907 (1977)
N. J. Gunther and P. G. L. Leach, Generalized invariants for the time-dependent harmonic oscillator, J. Math. Phys. 18, 572–576 (1977)
W. Sarlet and L. Y. Bahar, A direct construction of first integrals for certain non-linear dynamical systems, Internat. J. Non-Linear Mech. 15, 133–146 (1980)
L. Cveticanin, Adiabatic invariants of dynamical systems with two degrees of freedom, Internat. J. Non-linear Mech. 29, 799–808 (1994)
B. D. Vujanovic, Conservation laws of rheo-linear dynamical systems with one- and two-degrees-of-freedom, Internat. J. Non-Linear Mech. 27, 309–322 (1992)
B. D. Vujanovic and S. E. Jones, Variational Methods in Nonconservative Phenomena, Academic Press, New York, 1989, p. 370
L. Cveticanin, Approximate analytical solutions to a class of non-linear equations with complex functions, J. Sound Vibration 157, no. 2, 289–302 (1992)
N. N. Bogolubov and Ju. A. Mitropolski, Asimptoticheskie metodi v teorii nelinejnih kolebanij, Gos. Fiz. Mat. Lit., Moscow, 1963
L. Cveticanin, An approximate solution of a coupled differential equation with variable parameter, Trans. ASME Ser. E. J. Appl. Mech. 60, no. 1, 214–217 (1993)
S. Bravo Yuste and J. Diaz Bejarano, Construction of approximate analytical solutions to a new class of nonlinear oscillator equations, J. Sound Vibration 110, 347–350 (1986)
S. Bravo Yuste and J. Diaz Bejarano, Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions, J. Sound Vibration 139, 151–163 (1990)
V. T. Coppola and R. H. Rand, Averaging using elliptic functions: Approximation of limit cycles, Acta Mechanica 81, 125–142 (1990)
S. Bravo Yuste, Quasi-pure-cubic oscillators studied using a Krylov-Bogoliubov method, J. Sound Vibration 158, 267–275 (1992)
S. Bravo Yuste, On Duffing oscillators with slowly varying parameters, Internat J. Non-Linear Mech. 26, 671–677 (1991)
L. Cveticanin, An approximate solution for a system of two coupled differential equations, J. Sound Vibration 152, 375–380 (1992)
M. Abramowitz and I. A. Stegun, Spravochnik po specialynyim funkcijam, Moscow, Nauka, 1979
Dj. S. Djukic, Adiabatic invariants for dynamical systems with one degree of freedom, Internat. J. Non-Linear Mech. 16, 489–498 (1981)
P. G. L. Leach, Invariants and wave functions for some time-dependent harmonic oscillator-type Hamiltonians, J. Math. Phys. 18, 1902–1907 (1977)
N. J. Gunther and P. G. L. Leach, Generalized invariants for the time-dependent harmonic oscillator, J. Math. Phys. 18, 572–576 (1977)
W. Sarlet and L. Y. Bahar, A direct construction of first integrals for certain non-linear dynamical systems, Internat. J. Non-Linear Mech. 15, 133–146 (1980)
L. Cveticanin, Adiabatic invariants of dynamical systems with two degrees of freedom, Internat. J. Non-linear Mech. 29, 799–808 (1994)
B. D. Vujanovic, Conservation laws of rheo-linear dynamical systems with one- and two-degrees-of-freedom, Internat. J. Non-Linear Mech. 27, 309–322 (1992)
B. D. Vujanovic and S. E. Jones, Variational Methods in Nonconservative Phenomena, Academic Press, New York, 1989, p. 370
L. Cveticanin, Approximate analytical solutions to a class of non-linear equations with complex functions, J. Sound Vibration 157, no. 2, 289–302 (1992)
N. N. Bogolubov and Ju. A. Mitropolski, Asimptoticheskie metodi v teorii nelinejnih kolebanij, Gos. Fiz. Mat. Lit., Moscow, 1963
L. Cveticanin, An approximate solution of a coupled differential equation with variable parameter, Trans. ASME Ser. E. J. Appl. Mech. 60, no. 1, 214–217 (1993)
S. Bravo Yuste and J. Diaz Bejarano, Construction of approximate analytical solutions to a new class of nonlinear oscillator equations, J. Sound Vibration 110, 347–350 (1986)
S. Bravo Yuste and J. Diaz Bejarano, Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions, J. Sound Vibration 139, 151–163 (1990)
V. T. Coppola and R. H. Rand, Averaging using elliptic functions: Approximation of limit cycles, Acta Mechanica 81, 125–142 (1990)
S. Bravo Yuste, Quasi-pure-cubic oscillators studied using a Krylov-Bogoliubov method, J. Sound Vibration 158, 267–275 (1992)
S. Bravo Yuste, On Duffing oscillators with slowly varying parameters, Internat J. Non-Linear Mech. 26, 671–677 (1991)
L. Cveticanin, An approximate solution for a system of two coupled differential equations, J. Sound Vibration 152, 375–380 (1992)
M. Abramowitz and I. A. Stegun, Spravochnik po specialynyim funkcijam, Moscow, Nauka, 1979
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Article copyright:
© Copyright 1996
American Mathematical Society
