Poisson structures and integrable systems

Stanisław P. Kasperczuk

Physica A: Statistical Mechanics and its Applications, 2000, vol. 284, issue 1, 113-123

Abstract: A Poisson coalgebra (Fun(N),Δ) is used to construct integrable Hamiltonian systems. We consider a Poisson structure given by the bivector P=Pab(x)(∂/∂xa)∧(∂/∂xb),x∈R3, which does not form a Lie algebra with respect to the Poisson bracket {xa,xb}P=Pab(x). We prove that this coalgebra may be used to generate integrable Hamiltonian systems. As an example we give the Poisson tensor P=νx3(∂/∂x2)∧(∂/∂x2)+νx2(∂/∂x3)∧(∂/∂x1)−(ν/2)(∂/∂x2)∧(∂/∂x3) and we show that it is linked with the Calogero system H(n)=λn∑i=1npi+μ∑i,k=1npipjcosν(qi−qj).

Keywords: Poisson manifolds; Poisson coalgebras; Casimir functions; Integrable systems; Calogero system (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:284:y:2000:i:1:p:113-123

DOI: 10.1016/S0378-4371(00)00234-X

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