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I think for time reversibility we need to see if we can predict velocity and position at time $t$ if the values at $t+\Delta t$ is known. Verlet Algorithm The integrator equation is given as: $r_{t+\Delta t} = 2r_t -r_{t-\Delta t} + \Delta t^2 a_t$ Given we know the positions at $t-\Delta t$ and $t$ we can predict the position at $t+\Delta t$. So if this is ...
Note the identity for the modified Bessel functions of the first kind, $e^z = I_0(z) + 2 \sum_{k=1}^{\infty} I_k(z)$ (Abramowitz and Stegun, Eq. 9.6.34, https://personal.math.ubc.ca/~cbm/aands/page_376.htm) Using it, we can rewrite the infinite sum as $\sum_{i=1}^{\infty} I_{\nu+i}(2\lambda) = \sum_{i=1}^{\infty} I_{i}(2\lambda) - \sum_{i=1}^{\nu} I_{i}(2\... 2 Let$A_s[1,\cdots,m]$be an array of$m$samples, where say$A_s[i].\text{time}$gives us the time sample$i$was taken at, and where we assume the items are sorted in ascending order of the time they were taken. Define$I \in \mathbb{R}$as the time interval value to space new samples, and define$T_f\$ as a final time where you do not want any equal spaced ...