New four-dimensional solutions of the Jacobi equations for Poisson structures

Abstract

A new four-dimensional family of skew-symmetric solutions of the Jacobi equations for Poisson structures is characterized. As a consequence, previously known types of Poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of the new family of solutions. Additionally, it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions, such as their Casimir invariants, symplectic structure and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a limited sample of known finite-dimensional Poisson structures. Moreover, the results developed are valid globally in phase space, thus ameliorating the usual scope of Darboux theorem which is of local nature.

References in corpus (9)

• Multiple Hamiltonian Structures for Toda-type systems
• Simple evaluation of Casimir invariants in finite-dimensional Poisson systems
• Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems
• New solutions of the Jacobi equations for three-dimensional Poisson structures
• A constant of motion in 3D implies a local generalized Hamiltonian structure
• Separation of variables in the Jacobi identities
• One solution of the 3D Jacobi identities allows determining an infinity of them
• Local Stability and Lyapunov Functionals for n-Dimensional Quasipolynomial Conservative Systems
• Computing Casimir invariants from Pfaffian systems

Cited by in corpus (5)

• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
• New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis
• Characterization, global analysis and integrability of a family of Poisson structures
• Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
• New global solutions of the Jacobi partial differential equations