New solution family of the Jacobi equations: Characterization, invariants, and global Darboux analysis

Abstract

A new family of skew-symmetric solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is characterized and analyzed. Such family has some remarkable properties. In first place, it is defined for arbitrary values of the dimension and the rank. Secondly, it is described in terms of arbitrary differentiable functions, namely it is not limited to a given degree of nonlinearity. Additionally, it is possible to determine explicitly the fundamental properties of those solutions, such as their Casimir invariants and the algorithm for the reduction to the Darboux canonical form, which have been reported only for a very limited sample of finite-dimensional Poisson structures. Moreover, such analysis is carried out globally in phase space, thus improving the usual local scope of Darboux theorem.

arXiv commentarXiv admin note: text overlap with arXiv:1910.04569

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Cited by in corpus (9)

• Bi-Hamiltonian Structures of Chaotic Dynamical Systems in 3D
• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
• On Time-dependent Hamiltonian Realizations of Planar and Nonplanar Systems
• Characterization, global analysis and integrability of a family of Poisson structures
• An integrable family of Poisson systems: Characterization and global analysis
• On the quest for generalized Hamiltonian descriptions of $3D$-flows generated by curl of a vector potential
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• Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
• Congruence method for global Darboux reduction of finite-dimensional Poisson systems