W Sarlet and F Vermeire 2004 J. Phys. A: Math. Gen. 37 6319 doi:10.1088/0305-4470/37/24/010
A class of Poisson–Nijenhuis structures on a tangent bundle
W Sarlet and F Vermeire
Show affiliationsEquipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson–Nijenhuis structure from a given type (1, 1) tensor field J on Q. It is argued that the complete lift Jc of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to T*Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case when Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.
- PACS
- MSC
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37J35 Completely integrable systems, topological structure of phase space, integration methods
53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (See also 58J60)
- Subjects
- Dates
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Issue 24 (18 June 2004)
Received 21 January 2004
Published 2 June 2004
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A class of Poisson–Nijenhuis structures on a tangent bundle
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