<s>
The	O
method	B-Algorithm
is	O
known	O
by	O
different	O
names	O
in	O
different	O
disciplines	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
similar	O
to	O
the	O
velocity	O
Verlet	B-Language
method	I-Language
,	O
which	O
is	O
a	O
variant	O
of	O
Verlet	B-Language
integration	I-Language
.	O
</s>
<s>
Leapfrog	B-Algorithm
integration	I-Algorithm
is	O
equivalent	O
to	O
updating	O
positions	O
and	O
velocities	O
at	O
different	O
interleaved	O
time	O
points	O
,	O
staggered	O
in	O
such	O
a	O
way	O
that	O
they	O
"	O
leapfrog	O
"	O
over	O
each	O
other	O
.	O
</s>
<s>
Leapfrog	B-Algorithm
integration	I-Algorithm
is	O
a	O
second-order	O
method	B-Algorithm
,	O
in	O
contrast	O
to	O
Euler	B-Algorithm
integration	I-Algorithm
,	O
which	O
is	O
only	O
first-order	O
,	O
yet	O
requires	O
the	O
same	O
number	O
of	O
function	O
evaluations	O
per	O
step	O
.	O
</s>
<s>
Unlike	O
Euler	B-Algorithm
integration	I-Algorithm
,	O
it	O
is	O
stable	O
for	O
oscillatory	O
motion	O
,	O
as	O
long	O
as	O
the	O
time-step	O
is	O
constant	O
,	O
and	O
.	O
</s>
<s>
One	O
use	O
of	O
this	O
equation	O
is	O
in	O
gravity	O
simulations	O
,	O
since	O
in	O
that	O
case	O
the	O
acceleration	O
depends	O
only	O
on	O
the	O
positions	O
of	O
the	O
gravitating	O
masses	O
(	O
and	O
not	O
on	O
their	O
velocities	O
)	O
,	O
although	O
higher-order	O
integrators	O
(	O
such	O
as	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
)	O
are	O
more	O
frequently	O
used	O
.	O
</s>
<s>
There	O
are	O
two	O
primary	O
strengths	O
to	O
leapfrog	B-Algorithm
integration	I-Algorithm
when	O
applied	O
to	O
mechanics	O
problems	O
.	O
</s>
<s>
The	O
first	O
is	O
the	O
time-reversibility	O
of	O
the	O
Leapfrog	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
second	O
strength	O
is	O
its	O
symplectic	B-Algorithm
nature	O
,	O
which	O
sometimes	O
allows	O
for	O
the	O
conservation	O
of	O
a	O
(	O
slightly	O
modified	O
)	O
energy	O
of	O
a	O
dynamical	O
system	O
(	O
only	O
true	O
for	O
certain	O
simple	O
systems	O
)	O
.	O
</s>
<s>
This	O
is	O
especially	O
useful	O
when	O
computing	O
orbital	O
dynamics	O
,	O
as	O
many	O
other	O
integration	O
schemes	O
,	O
such	O
as	O
the	O
(	O
order-4	O
)	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
method	I-Algorithm
,	O
do	O
not	O
conserve	O
energy	O
and	O
allow	O
the	O
system	O
to	O
drift	O
substantially	O
over	O
time	O
.	O
</s>
<s>
Because	O
of	O
its	O
time-reversibility	O
,	O
and	O
because	O
it	O
is	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
,	O
leapfrog	B-Algorithm
integration	I-Algorithm
is	O
also	O
used	O
in	O
Hamiltonian	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
,	O
a	O
method	B-Algorithm
for	O
drawing	O
random	O
samples	O
from	O
a	O
probability	O
distribution	O
whose	O
overall	O
normalization	O
is	O
unknown	O
.	O
</s>
