I am a materials science undergrad, interested to know the algorithms for numerical integration of equation of motion in computational materials science, like molecular dynamics. It is said that, time-reversal symmetry is essential for such simulations, while classic integration schemes like Trapezoidal, simpsons or weddle methods handle previous and next time step differently. So verlet algorithm is used instead.
Position verlet indeed adds previous and past timesteps and maintains time-reversal symmetry. But velocity verlet doe not. Why is time-reversal symmetry not important for velocity? is it because time reversal symmetry is meaningful only for position and its even derivatives, as in newton's law of motion?
My knowledge on Numerical analysis is only of introductory level, and i have not deeply studied Lagrangian, chaos theory, group theory or hyper-dimensional geometry yet.
You have already stated, that position-verlet is time-reversible. By construction, a(t) is therefore also time-reversible. If you use this knowledge and transform your equations of motions accordingly, you will see, that v(t) is also time-reversible.
Both satisfy the time-reversibility because the discrete flow map, F(dt), is time reversible:
F(dt) = exp(i L1 dt/2) exp(i dt L2) exp(i L1 dt/2)
where dt is the time step and
iL1 = a(t) (\partial / \partial v)
iL2 = v(t) (\partial / \partial x)
a = acceleration
v = velocity
x = position
Applying this operator (F(dt)), the velocity-Verlet algorithm is obtained. Now, check it
F(dt)F(-dt) = 1
If we take
F(dt) = exp(i L2 dt/2) exp(i dt L1) exp(i L2 dt/2)
Then, the position Verlet algorithm is obtained, which is time-reversible too, F(dt)F(-dt)=1.
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