Combinatorics of Hamiltonian Normal Forms

TL;DR

This paper explores algebraic and combinatorial methods to describe Hamiltonian normal forms directly from the original Hamiltonian, providing explicit formulas without normalization procedures.

Contribution

It introduces a novel approach to compute Hamiltonian normal forms explicitly in terms of the original Hamiltonian, applicable in arbitrary dimensions.

Findings

Derived explicit nonlinear functional formulas for normal forms in one degree of freedom.

Extended the formulas to higher dimensions, maintaining explicitness.

Provided a combinatorial framework for understanding the normal form structure.

Abstract

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. We present analogous results in arbitrary dimension. The corresponding formulas are more complicated but still explicit.