 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
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| A. G. Petrov, "On averaging Hamiltonian systems," Mech. Solids. 36 (3), 13-25 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
3 |
Pages |
13-25 |
| Title |
On averaging Hamiltonian systems |
| Author(s) |
A. G. Petrov (Moscow) |
| Abstract |
The Cauchy problem is considered for Hamiltonian equations in standard form,
with the Hamiltonian being a periodic function of time.
On the basis of the Hamiltonian, a mapping is constructed for
the determination of
the Poincaré first return points. This mapping is expressed
in terms of a single function defined by an equation of Hamilton-Jacobi type.
The solution is obtained as a standard convergent expansion in powers of
a small parameter. Partial sums of the series determine
a sequence that converges to the exact Poincaré mapping for the Hamiltonian system.
The mapping corresponding to each approximation
has its Jacobian identically equal to unity.
The approach developed here is applied for the construction of
a new averaging procedure for Hamiltonian systems.
An autonomous Hamiltonian approximating the Poincaré first return points
is represented as an asymptotic series. As a demonstration,
the method is applied to the following problems:
(i) motion of particles of an incompressible fluid in a thin layer
whose boundary varies in a periodic way; (ii) motion of a spherical pendulum
with arbitrarily oscillating pivotal point. |
| References |
| 1. | N. N. Bogolyubov and Yu. A. Mitropol'skii,
Asymptotic Methods in the Theory of Nonlinear Vibrations [in Russian],
Nauka, Moscow, 1974. |
| 2. | V. Ph. Zhuravlev and D. M. Klimov,
Applied Methods in Vibration Theory [in Russian], Nauka, Moscow, 1988. |
| 3. | A. I. Neishtadt,
"On the separation of motions in systems with a rapidly rotating phase,"
PMM [Applied Mathematics and Mechanics], Vol. 48, No. 3, pp. 197-204, 1984. |
| 4. | V. I. Arnold, Mathematical Methods in Classical Mechanics [in Russian],
Nauka, Moscow, 1974. |
| 5. | J. Moser, Lectures on Hamiltonian Systems [Russian translation], Mir,
Moscow, 1973. |
| 6. | A. P. Markeev, Libration Points in Celestial Mechanics and
Space Dynamics [in Russian], Nauka, Moscow, 1978. |
| 7. | V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt,
Mathematical Aspects of Classical and Celestial Mechanics.
Advances in Science and Technology. Modern Problems in Mathematics [in Russian],
VINITI, Moscow, 1985. |
| 8. | A. G. Petrov, "On averaging Hamiltonian systems,"
Doklady AN SSSR, Vol. 368, No. 4, pp. 483-488, 1999. |
| 9. | R. Courant and D. Hilbert,
Methods of Mathematical Physics. Volume 2 [Russian translation],
Gostekhizdat, Moscow, 1951. |
| 10. | G. E. O. Giacaglia,
Perturbation Methods in Nonlinear Systems [Russian translation],
Nauka, Moscow, 1979. |
| 11. | A. G. Petrov, "On the motion of particles of
an incompressible medium," Izv. AN. MZhG
[Journal of Fluid Mechanics], No. 4, pp. 12-19, 2000. |
| 12. | L. N. Sretenskii, Theory of Wave Motion of Fluids [in Russian],
Nauka, Moscow, 1977. |
| 13. | O. M. Phillips, Dynamics of the Upper Ocean Layer [Russian
translation], Gidrometeoizdat, Leningrad, 1980. |
| 14. | L. D. Landau and E. M. Lifshitz, Theoretical Physics,
Vol. 1, Mechanics [in Russian], Nauka, Moscow, 1965. |
| 15. | B. S. Bardin and A. P. Markeev,
"On the stability of equilibrium of a pendulum with
vertically oscillating pivotal point," PMM [Applied Mathematics and
Mechanics], Vol. 59, No. 6, pp. 922-929, 1995. |
| 16. | L. D. Akulenko, "Asymptotic analysis of
dynamical systems
subjected to high-frequency action," PMM [Applied Mathematics
and Mechanics], Vol. 58, No. 3, pp. 23-31, 1994. |
| 17. | A. P. Markeev, "On the dynamics of a spherical pendulum
with vibrating pivotal point,"
PMM [Applied Mathematics
and Mechanics], Vol. 63, No. 2, pp. 210-213, 1999. |
|
| Received |
22 March 1999 |
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