Equipping the tangent bundle T Q 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 J c of J is not the natural candidate for a Nijenhuis tensor on T Q, 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.

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