Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 720–734 | Cite as

Poisson Brackets after Jacobi and Plücker

  • Pantelis A. DamianouEmail author


We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.


Poisson structures Plücker relations 

MSC2010 numbers

53D17 37J35 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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