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h2. A closer look: Velocity Verlet Algorithm

We designed a dynamical simulation to compute the particle trajectories as a function of time. The simulation utilizes the Velocity Verlet algorithm, which calculates positions and velocities of particles by means ofvia Taylor expansion. Because the Newton's equation of motion is second order in relative position (r), the initial condition needs to specify both particle position and velocity at time 0.

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The model makes use of the following equations are used:

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{latex}
\large
$$
{{x(t + \Delta t) }} = {{ x(t) + v(t)\Delta t + (1/2)a(t)\Delta t^2    }}
$$
{latex}
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{latex}
\large
$$
{{v(t + \Delta t/2) }} = {{ v(t) + (1/2)a(t)\Delta t    }}
$$
{latex}


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!fig2.png|align=center!