I have the following classical Hamiltonian for two coupled oscillators in the same molecule: $$H=T+V =\left(\frac{p_1^2}{2\mu}+\frac{p_2^2}{2\mu}+k_pp_1p_2\right)+ \left(\frac{1}{2}\mu\omega_1^{2}x_1^2+\frac{1}{2}\mu\omega_2^{2}x_2^2+k_xx_1x_2\right),$$ where $p_1$ and $p_2$ are momenta and $x_1$ and $x_2$ are displacements; $k_p$ and $k_x$ are constants; $\mu$ is the reduced mass; and $\omega$ is the resonant (harmonic) frequency. I need a velocity-Verlet-style integrator for this Hamiltonian. I know I can derive one from the Liouvillian acting on the phase space probability density $$\rho(t) = e^{iL}\rho(0),$$ but I would rather not go through that exercise if I don't have to. Does anyone know a source that reports an integrator algorithm for this type of Hamiltonian or one similar to it? The real-life situation is two chemical bonds (O-H) vibrating with a shared oxygen atom.
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