# Stable time step limits for Velocity-Verlet integration

I'm implementing a mass-spring solid mechanics solver and I'd like to use the Velocity-Verlet time integration scheme. However, I cannot find anything about the maximum stable time step -- either saying there is none and it's unconditionally stable like an implicit scheme or that there is one and it's bound by something like the explicit schemes.

I would think the maximum time step is related to the maximum force and the minimum distance between the particles if there is one, but I can't seem to find anything. What is the stable time step limit for the Velocity-Verlet integration?

At least for leapfrog integration, one can analytically calculate the maximum time step for quadratic Hamiltonians. For such systems, a leapfrog step is a linear mapping $(q(t), p(t) \mapsto (q(t+\epsilon), p(t+\epsilon))$. Stability then depends on the magnitude of the eigenvalues of the matrix representing this mapping. For a harmonic oscillator with force constant $k$, these eigenvalues are given by $(1 - \frac{k\epsilon^2}{2}) \pm \epsilon \sqrt k \sqrt{\frac{k \epsilon^2}{4} - 1}$. So for $\sqrt{k}{\epsilon} < 2$, the eigenvalues are complex and have a squared magnitude of $1$. The system simulated with this timestep will therefor be stable. On the other hand, if $\sqrt{k} \epsilon > 2$, the eigenvalues are real and at least one of them will have an absolute value $> 0$. This means a simulation with this time step will be unstable when you iterate this scheme.