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Zbl 1092.53059
Ortega, Juan-Pablo; Planas-Bielsa, Víctor
Dynamics on Leibniz manifolds. (English)
[J] J. Geom. Phys. 52, No. 1, 1-27 (2004). ISSN 0393-0440

With help of the Poisson tensor $P$ on a manifold $M$, one associates a Hamiltonian vector field (dynamics) $X_f=i_{df}P$ with every function $f$ on the Poisson manifold $(M,P)$. Then, the map $f\mapsto X_f$ is a Lie algebra homomorphism if one takes the Poisson bracket $\{ f,g\}_P=\langle P,df\wedge dg\rangle$ on the associative algebra of functions $C^\infty(M)$ on one hand, and the bracket of vector fields on the second. The Poisson bracket has the obvious Leibniz property: $\{ f,gh\}_P=\{ f,g\}_Ph+g\{ f,h\}_P$, which just tells us that $X_f=\{ f,\cdot\}_P$ is a vector field. \par If we do not assume that $P$ is Poisson, or even that it is a bi-vector field, so that $P$ is just a 2-contravariant tensor field on $M$, then the corresponding bracket of functions $\{ f,g\}_P$ is no longer Lie nor even skew-symmetric. What survives is only the Leibniz property which, however, should be understood for both arguments separately. This explains the name -- the Leibniz structure -- for such a tensor. \par The paper is devoted to dynamics associated with Leibniz structures. The gradient flows, some control and dissipative systems, and certain nonholonomic systems turn out to be examples of the Hamiltonian vector fields for Leibniz structures. Symmetries of such systems and the associated reduction procedures are described as well.
[Janusz Grabowski (Warszawa)]

MSC 2000:: *53D17 Poisson manifolds
17B63 Poisson algebras
17D99 Other nonassociative rings and algebras
37J05 Relations with symplectic geometry and topology
70G45 Differential-geometric methods

Keywords: Leibniz manifold; Poisson bracket; Hamiltonian mechanics; nonholonomic systems

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