Nonlinear Dynamics

, Volume 48, Issue 1–2, pp 185–197

# On the normal forms of Hamiltonian systems

• Jan Awrejcewicz
• Alexander G. Petrov
Original Article

## Abstract

We propose a novel method to analyze the dynamics of Hamiltonian systems with a periodically modulated Hamiltonian. The method is based on a special parametric form of the canonical transformation $${\bf q},{\bf p}\to {\bf Q},{\bf P}$$, $$H(t,{\bf q},{\bf p})\to \bar H(t,{\bf Q},{\bf P})$$ $$\displaylines{ \begin{array}{c} \left\{ \begin{array}{c} {\bf q}={\bf x}-\displaystyle\frac12\Psi_{\bf x} {\bf p}={\bf y}+\displaystyle\frac12\Psi_{\bf y} \end{array} \right., \qquad \left\{ \begin{array}{c} {\bf Q}={\bf x}+\displaystyle\frac12\Psi_{\bf x} {\bf P}={\bf y}-\displaystyle\frac12\Psi_{\bf y} \end{array} \right. \end{array}\cr \bar H(t,{\bf Q},{\bf P})=\Psi_t (t,{\bf x},{\bf y})+H(t,{\bf Q},{\bf P}) }$$ using Poincaré generating function Ψ (t,x,y). As a result, stability problem of a periodic solution is reduced to finding a minimum of the Poincaré function.

The proposed method can be used to find normal forms of Hamiltonians. It should be emphasized that we apply the modified concept of Zhuravlev [Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997); Prikladnaya Matematika i Mekhanika 66(3), (2002) in Russian] to define an invariant normal form, which does not require any partition to either autonomous – non-autonomous, or resonance – non-resonance cases, but it is treated in the frame of one approach. In order to find the corresponding normal form asymptotics, a system of equations is derived analogous to Zhuravlev's chain of equations. Instead of the generator method and guiding Hamiltonian, a parametrized guiding function is used. It enables a direct (without the transformation to an autonomous system as in Zhuravlev's method) computation of the chain of equations for non-autonomous Hamiltonians. For autonomous systems, the methods of computation of normal forms coincide in the first and second approximations.

Using this method we will present solutions of the following problems: nonlinear Duffing oscillator; oscillation of a swinging spring; dynamics of solid particles in the acoustic wave of viscous liquid, and other problems.

## Keywords

Hamiltonian mechanics Normal form Swinging oscillator

## References

1. 1.
Petrov, A.G.: A parametric method of constructing Poincaré mappings in hydrodynamic systems. J. Appl. Math. Mech. 66(6), 905–921 (2002)
2. 2.
Awrejcewicz, J., Krysko, V.A.: Introduction to Asymptotic Methods. Chapman and Hall, CRC Press, Boca Raton, London, New York (2006)
3. 3.
Verhulst, F.: Methods and Applications of Singular Perturbations. Texts in Applied Mathematics, Springer Verlag, New York (2005)Google Scholar
4. 4.
Verhulst, F.: Differential Equations and Dynamical Systems, 2nd Edition. Springer Verlag, Berlin (2006)Google Scholar
5. 5.
Petrov, A.G.: Invariant normalization of non-autonomous Hamiltonian systems. J. Appl. Math. Mech. 68, 357–367 (2004)
6. 6.
Petrov, A.G.: Asymptotic methods for solving the hamilton equations with the use of a parametrization of canonical transformations. J. Diff. Equ. 40(5), 672–685 (2004)
7. 7.
Poincaré, A.: Collected Works in Three Volumes, vol. II. Nauka, Moscow (1972) in RussianGoogle Scholar
8. 8.
Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd Edition. Graduate Text in Mathematics, Springer, Berlin (1989)Google Scholar
9. 9.
Petrov, A.G.: On averaging the Hamiltonian system. Mech. Solids 36(3), 13–25 (2001), Izvestia AN MTT (3), in RussianGoogle Scholar
10. 10.
Zhuravlev, V.F.: Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997)Google Scholar
11. 11.
Zhuravlev, V.F.: Invariant normalization of non-automomous Hamiltonians. Prikladnaya Matematika i Mekhanika 66(3), (2002) in RussianGoogle Scholar
12. 12.
Petrov, A.G.: Modification of the method of invariant normalization of hamiltonians by parameterizing canonical transformations. Dokl. Phys. 47, 742 (2002)
13. 13.
Birkhoff, G.D.: Dynamical Systems. American Mathematical Society, New York (1927
14. 14.
Bruno, A.D.: The Restricted Three Body Three Problem: Periodic Orbits. Nauka, Moscow (1990) in Russian
15. 15.
Arnold, V.I.: Additional Problems of the Theory of Ordinary Differential Equations. Nauka, Moscow (1978) in RussianGoogle Scholar
16. 16.
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspect of classical and universe mechanics. Encyclopedia of Mathematical Sciences, vol. 3. Springer-Verlag, Berlin (1993)Google Scholar
17. 17.
Ganiyev, R.F., Ukrainskiy, L.E.: Dynamics of Particles Subject to Vibrations. Naukova Dumka, Kiev (1975) in RussianGoogle Scholar
18. 18.
Nigmatulin, R.I.: Dynamics of Multi-plate Media. Nauka, Moscow (1987) in RussianGoogle Scholar
19. 19.
Bogoliubov, N.N., Mitropolskii, Yu.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach, New York (1961)Google Scholar
20. 20.
Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)
21. 21.
Starginsky, V.M.: Applied Methods of Nonlinear Oscillations. Nauka, Moscow (1977) in RussianGoogle Scholar
22. 22.
Bogaevskiy, V.N., Povzner, A.Ya.: Algebraic Methods in nonlinear theory of Perturbation. Nauka, Moscow (1987) in RussianGoogle Scholar
23. 23.
Petrov, A.G., Zaripov, M.N.: Nonlinear oscillations of a swinging spring. Dokl. Akademii Nauk, Mekhanika 399(3), 347–352 (2004) in Russian
24. 24.
Klimov, D.M., Leonov, V.V., Rudenko, V.M.: Symbolic computation methods in nonlinear tasks of mechanics. Mech. Solids 39(3), 1986, Izvestia AN MTT (6), in RussianGoogle Scholar