  An outstanding property of any Hamiltonian system is the symplecticity of its
flow, namely, the continuous trajectory preserves volume in phase space. Given
a symplectic but discrete trajectory generated by a transition matrix applied
at a fixed time-increment ($\tau > 0$), it was generally believed that there
exists a unique Hamiltonian producing a continuous trajectory that coincides at
all discrete times ($t = n\tau$ with $n$ integers) as long as $\tau$ is small
enough. However, it is now exactly demonstrated that, for any given discrete
symplectic dynamics of a harmonic oscillator, there exist an infinite number of
real-valued Hamiltonians for any small value of $\tau$ and an infinite number
of complex-valued Hamiltonians for any large value of $\tau$. In addition, when
the transition matrix is similar to a Jordan normal form with the supradiagonal
element of $1$ and the two identical diagonal elements of either $1$ or $-1$,
only one solution to the Hamiltonian is found for the case with the diagonal
elements of $1$, but no solution can be found for the other case.
