...
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 via 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.
The model makes use of the following equations:
Latex |
---|
0.
\\
{latex}
\large
$$
{{x(t + \Delta t) }} = {{ x(t) + v(t)\Delta t + (1/2)a(t)\Delta t^2 }}
$$
|
Latex |
---|
{latex} \\ {latex} \large $$ {{v(t + \Delta t/2) }} = {{ v(t) + (1/2)a(t)\Delta t }} $$ {latex} \\ !fig2.png|align=center! |
with the given time step and initial value conditions.
The whole procedure can be summarized in following steps: