Non-unique Hamiltonians for Discrete Symplectic Dynamics

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.