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The authors study some geometrical constructions on the cotangent bundle $T^*M$ from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on $T^*M$ into vector fields on $T^*M$ are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of $T^*M$ and by the Liouville vector field of $T^*M$. Then they determine all natural operators transforming pairs of functions on $T^*M$ into functions on $T^*M$. In this case, the main generator is the classical Poisson bracket.
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