Exact integration of Hamiltonian dynamics via Jacobi and Poisson Cinf-structures

TL;DR

This paper introduces a geometric framework for the exact integration of Hamiltonian systems using Poisson and Jacobi structures, enabling explicit solutions even without complete conserved quantities.

Contribution

It develops a novel Poisson $C^ abla$-structure approach for integrating Hamiltonian systems, extending to Jacobi systems and providing explicit schemes without requiring full integrability.

Findings

Applied to Toda lattice and Vlasov equation reductions.

Provides explicit integration schemes for non-integrable systems.

Extends to time-dependent Hamiltonian systems.

Abstract

We develop a geometric framework for the exact integration of Hamiltonian systems based on triangular closure relations among a finite family of functions. Unlike Liouville-Arnold integrability and its noncommutative generalizations, the functions involved in these relations need not be first integrals of the system. Instead, their Hamiltonian vector fields generate a $C^\infty$-structure on phase space that provides an algorithmic procedure for integrating the dynamics. Within this framework, the equations of motion can be reduced to a finite sequence of completely integrable Pfaffian equations, yielding an explicit integration scheme even when a complete set of conserved quantities is unavailable. The resulting geometric structure is called a Poisson $C^\infty$-structure. We further extend the construction to Jacobi Hamiltonian systems, showing that the same mechanism applies naturally to important subclasses of Jacobi geometry, including Poisson, locally conformally symplectic, and contact manifolds. The method is illustrated on two systems of physical interest: the two-particle non-periodic Toda lattice and the multi-waterbag reduction of the Vlasov equation. We also discuss extensions of the theory to time-dependent Hamiltonian systems.