Algorithms for Computing Hamiltonian Normal Form

The invariant normalization method proposed by V.F. Zhuravlev, used for calculating normal or symmetrized forms of autonomous Hamiltonian systems, is discussed. The normalizing canonical transformation is represented by a Lie series using a generating Hamiltonian. This method has a generalization proposed by A.G. Petrov, which normalizes not only autonomous but also nonautonomous Hamiltonian systems. The normalizing canonical transformation is represented by a series using a parametric function. For autonomous Hamiltonian systems, the first two approximation steps in both methods are the same, and the remaining steps are different. The normal forms of both methods are identical. A method for testing a normalization program has also been proposed. For this purpose, the Hamiltonian of a strongly nonlinear Hamiltonian system is found, for which the normal form is a quadratic Hamiltonian. The normalizing transformation is expressed in terms of elementary functions.