Are all 1 dimensional Poisson manifold trivial?

Question

The title says it all. How to rigorously prove or disprove that all poisson brackets on C^{inf}(R) are trivial ?

Answer 1

The Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. On a $1$-dimensional vector space, the Lie bracket is trivial, because $$ \{x,y\}=-\{y,x\}, $$ so that it follows that $\{\lambda x,\mu x\}=\lambda \mu \{x,x\}=0$.