Existence of Hamiltonian Structure in 3D

TL;DR

This paper investigates the conditions under which a bi-Hamiltonian structure exists in three-dimensional systems, linking solutions of Riccati equations to cohomology classes and providing explicit examples and obstructions to integrability.

Contribution

It establishes a connection between Riccati equations, cohomology classes, and Hamiltonian structures in 3D, offering explicit constructions and identifying obstructions.

Findings

Explicit solutions relate Riccati coefficients to conserved quantities.

Bi-Hamiltonian systems are fully characterized when certain cohomology classes vanish.

Godbillon-Vey invariant acts as an obstruction in the Darboux-Halphen system.

Abstract

In three dimensions, the construction of bi-Hamiltonian structure can be reduced to the solutions of a Riccati equation with the arclength coordinate of a Frenet-Serret frame being the independent variable. Explicit integration of conserved quantities are connected with the coefficients of Riccati equation which are elements of the third cohomology class. All explicitly constructed examples of bi-Hamiltonian systems are exhausted when this class along with the first one vanishes. The latter condition provides integrating factor for explicit integration of Hamiltonian functions. For the Darboux-Halphen system, the Godbillon-Vey invariant is shown to arise as obstruction to integrability of integrating factor.