Non-unique Hamiltonians for Discrete Symplectic Dynamics

TL;DR

This paper demonstrates that discrete symplectic dynamics, such as those of a harmonic oscillator, can correspond to infinitely many Hamiltonians, challenging the belief in uniqueness for small time steps and revealing complex solution structures.

Contribution

It proves the existence of multiple Hamiltonians for the same discrete symplectic trajectory, including real and complex solutions, for various transition matrices.

Findings

Infinite Hamiltonians exist for harmonic oscillator dynamics.

Uniqueness of Hamiltonian is not guaranteed for small or large time steps.

Certain matrix forms admit only one or no Hamiltonian solutions.

Abstract

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.