Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems
Rafael Azuaje, Piergiulio Tempesta
TL;DR
The paper develops Jacobi-Haantjes manifolds by merging Jacobi structures with Abelian extended Haantjes algebras to provide a unified notion of integrability for both conservative and dissipative Hamiltonian dynamics. It proves a Liouville-type result for contact Hamiltonian systems: complete integrability is equivalent to the existence of an Abelian extended Haantjes algebra, and dissipative flows lift to conservative flows on a Poisson-Haantjes manifold via symplectization, enabling separation of variables. A general SoV theory for dissipative systems is established through Darboux-Haantjes coordinates on the lifted space, which project to separation coordinates for the original dissipative dynamics. The authors construct new completely integrable dissipative systems in various dimensions (including 3D, 5D, and a general (2n+1)-dimensional family) and provide explicit Jacobi-Haantjes structures that govern their integrability and separation properties, thereby unifying conservative and dissipative integrability in a common geometric framework.
Abstract
The notion of Jacobi-Haantjes manifold, consisting of a Jacobi manifold endowed with an algebra of extended Haantjes operator fields, is proposed as a natural geometric framework which allows us to define the notion of integrability of both conservative and dissipative Hamiltonian systems, in a unified way. As a reduction, contact-Haantjes manifolds are defined. We prove that the integrability of a contact Hamiltonian system is equivalent to the existence of a suitable Abelian extended Haantjes algebra associated with the system. This result allows us to define a large class of new, completely integrable contact Hamiltonian systems from a given extended Haantjes algebra. Moreover, we propose a theory of separation of variables for dissipative systems. This result is achieved by lifting a dissipative system into a higher-dimensional manifold, obtained as the symplectization of the Jacobi-Haantjes structure associated with the system. This new manifold naturally acquires the structure of a symplectic-Haantjes manifold. We prove that the Darboux-Haantjes coordinates which separate the Hamilton-Jacobi equation of the higher-dimensional symplectic-Haantjes manifold are in fact separation variables for the Hamilton equations associated with the original dissipative system.