The inverse problem of the Birkhoff-Gustavson normalization and ANFER, Algorithm of Normal Form Expansion and Restoration
Yoshio Uwano
TL;DR
This work tackles the inverse problem of BG-normalization by developing ANFER, a computer-algebra–based procedure that uses a systematic composition of non-infinitesimal canonical transformations to recover Hamiltonians sharing a given BG-normal form $G(\xi,\eta)$. The paper establishes a rigorous degree-$\rho$ framework, detailing image/ker decompositions and Stage-based recursions that render the inverse problem tractable and memory-efficient. It then applies ANFER to BDIC-related questions for PHOCPs and PHOQPs, deriving explicit degree-4 results and showing that BDIC-enabled equivalence of BG-normal forms implies integrability and separability in these oscillator families. The approach provides a practical method to generate and analyze new integrable families, with potential implications for separability and quantum bifurcation studies, and is implemented in REDUCE with available prototype code. Overall, the work advances both the theory and computation of the inverse BG-normalization and demonstrates concrete BDIC-driven insights for nonlinear oscillators.
Abstract
In the series of papers [1-4], the inverse problem of the Birkhoff-Gustavson normalization was posed and studied. To solve the inverse problem, the symbolic-computing program named ANFER (Algorithm of Normal Form Expansion and Restoration) is written up, with which a new aspect of the Bertrand and Darboux integrability condition is found \cite{Uwano2000}. In this paper, the procedure in ANFER is presented in mathematical terminology, which is organized on the basis of the composition of canonical transformations.