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