  In two recent papers [N. Aizawa, Y. Kimura, J. Segar, J. Phys. A 46 (2013)
405204] and [N. Aizawa, Z. Kuznetsova, F. Toppan, J. Math. Phys. 56 (2015)
031701], representation theory of the centrally extended l-conformal Galilei
algebra with half-integer l has been applied so as to construct second order
differential equations exhibiting the corresponding group as kinematical
symmetry. It was suggested to treat them as the Schrodinger equations which
involve Hamiltonians describing dynamical systems without higher derivatives.
The Hamiltonians possess two unusual features, however. First, they involve the
standard kinetic term only for one degree of freedom, while the remaining
variables provide contributions linear in momenta. This is typical for
Ostrogradsky's canonical approach to the description of higher derivative
systems. Second, the Hamiltonian in the second paper is not Hermitian in the
conventional sense. In this work, we study the classical limit of the quantum
Hamiltonians and demonstrate that the first of them is equivalent to the
Hamiltonian describing free higher derivative nonrelativistic particles, while
the second can be linked to the Pais-Uhlenbeck oscillator whose frequencies
form the arithmetic sequence omega_k=(2k-1), k=1,...,n. We also confront the
higher derivative models with a genuine second order system constructed in our
recent work [A. Galajinsky, I. Masterov, Nucl. Phys. B 866 (2013) 212] which is
discussed in detail for l=3/2.
