  We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian
of a fourth-order differential equation. In particular we consider the
classical problem of the Pais-Uhlenbeck oscillator and write down the
accompanying Hamiltonian. We then compare such an expression with an
alternative derivation of the Hamiltonian that makes use of the Ostrogradski's
method and show that a mapping from the one to the other is achievable by
variable transformations. Assuming canonical quantization procedure to be valid
we go for the operator version of the Hamiltonian that represents a pair of
uncoupled oscillators. This motivates us to propose a generalized
Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation
and annihilation operators by including terms quadratic and linear in them.
Such a Hamiltonian turns out to be essentially non-Hermitian but has an
equivalent Hermitian representation which is reducible to a typically
position-dependent reduced mass form.
