Nonlinear Dynamics

, Volume 48, Issue 1–2, pp 185–197 | Cite as

On the normal forms of Hamiltonian systems

  • Jan Awrejcewicz
  • Alexander G. Petrov
Original Article


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.


Hamiltonian mechanics Normal form Swinging oscillator 


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Copyright information

© Springer Science + Business Media, B.V. 2006

Authors and Affiliations

  1. 1.Department of Automatics and BiomechanicsTechnical University of LodzLodzPoland
  2. 2.Russian Academy of SciencesInstitute for Problems in MechanicsMoscowRussia

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