Liyan Ni | Searcharxiv

arXiv2024

On the relation between the velocity- and position-Verlet integrators

The difference and similarity between the velocity- and position-Verlet integrators are discussed from the viewpoint of their Hamiltonian representations for both linear and nonlinear systems. For a harmonic oscillator, the exact Hamiltonians reveal that positional trajectories generated by the two integrators follow an identical second-order differential equation and thus can be matched by adjusting initial conditions. In contrast, the series expansion of the Hamiltonians for the nonlinear discrete dynamics clearly indicate that the two integrators differ fundamentally. These analytical results are confirmed by simple numerical simulations of harmonic and anharmonic oscillators.

physics.comp-ph↗

arXiv2024

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.

math-ph↗