Solvable Structures for Hamiltonian Systems
Sasa Kresic-Juric, Concepcion Muriel, Adrian Ruiz
TL;DR
The paper develops a canonical solvable-structure framework for completely integrable Hamiltonian systems, enabling explicit solution by quadratures. By constructing a solvable chain with $A=\partial_t+X_H$, first integrals $F_i$, and momentum-dependent $G_i$, it derives Pfaffian forms whose integration yields action variables $P_i$ (top forms) and angle variables $Q_i$ (bottom forms), providing a constructive reinterpretation of the Arnold–Liouville theorem. The approach is demonstrated on the direct sum of $n$ harmonic oscillators and the rational Calogero–Moser system (N=2), producing explicit solutions and clear action–angle variables. This framework offers a systematic, symmetry-based route to integration by quadratures and a practical method to obtain action–angle coordinates in integrable Hamiltonian dynamics.
Abstract
In this paper, we investigate solvable structures associated to Hamiltonian equations. For a completely integrable Hamiltonian system with $n$ degrees of freedom, we construct a canonical solvable structure consisting of $2n$ Hamiltonian vector fields. We derive explicit expressions for the corresponding Pfaffian forms, whose integration provides solutions to the Hamiltonian equations. We show that the upper $n$ forms give the action varibles, while the lower $n$ forms yield the angle variables of the system. This offers a novel interpretation of the Arnold--Liouville theorem in terms of solvable structures. We ilustrate the theory by deriving explicit solutions and action--angle variables for $n$ harmonic oscillators and the Calogero--Moser system.