  Newton-Hooke group is the nonrelativistic limit of de Sitter (anti-de Sitter)
group, which can be enlarged with transformations that describe constant
acceleration, as well as central charges. We consider a higher order Lagrangian
that is quasi-invariant under the acceleration-extended Newton-Hooke symmetry,
and obtain the Schr\"{o}dinger equation quantizing the Hamiltonian
corresponding to its first order form. We show that the Schr\"{o}dinger
equation is invariant under the acceleration-extended Newton-Hooke
transformations. We also discuss briefly the exotic conformal Newton-Hooke
symmetry in 2+1 dimension.
