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However, later on, he uses the lie derivative of the hamiltonian, which is a scalar function. I.e. L(H)f () = subscript

I'm assuming that this means the vector field induced by the hamiltonian, and not the lie derivative in the direction of the hamiltonian itself. However, usually Thirring is careful to call out this distinction (i.e. notating X(H) as the vector field induced by the hamiltonian).

My question: is it possible to have a lie derivative in the direction of a scalar function?

thanks