<s>
In	O
mathematics	O
,	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
(	O
SI	O
)	O
is	O
a	O
numerical	B-Algorithm
integration	I-Algorithm
scheme	I-Algorithm
for	O
Hamiltonian	O
systems	O
.	O
</s>
<s>
Symplectic	B-Algorithm
integrators	I-Algorithm
form	O
the	O
subclass	O
of	O
geometric	B-Algorithm
integrators	I-Algorithm
which	O
,	O
by	O
definition	O
,	O
are	O
canonical	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
They	O
are	O
widely	O
used	O
in	O
nonlinear	O
dynamics	O
,	O
molecular	O
dynamics	O
,	O
discrete	B-Algorithm
element	I-Algorithm
methods	I-Algorithm
,	O
accelerator	O
physics	O
,	O
plasma	O
physics	O
,	O
quantum	O
physics	O
,	O
and	O
celestial	O
mechanics	O
.	O
</s>
<s>
A	O
numerical	O
scheme	O
is	O
a	O
symplectic	B-Algorithm
integrator	I-Algorithm
if	O
it	O
also	O
conserves	O
this	O
2-form	O
.	O
</s>
<s>
Symplectic	B-Algorithm
integrators	I-Algorithm
also	O
might	O
possess	O
,	O
as	O
a	O
conserved	O
quantity	O
,	O
a	O
Hamiltonian	O
which	O
is	O
slightly	O
perturbed	O
from	O
the	O
original	O
one	O
(	O
only	O
true	O
for	O
a	O
small	O
class	O
of	O
simple	O
cases	O
)	O
.	O
</s>
<s>
Most	O
of	O
the	O
usual	O
numerical	O
methods	O
,	O
like	O
the	O
primitive	O
Euler	B-Algorithm
scheme	I-Algorithm
and	O
the	O
classical	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
scheme	I-Algorithm
,	O
are	O
not	O
symplectic	B-Algorithm
integrators	I-Algorithm
.	O
</s>
<s>
A	O
widely	O
used	O
class	O
of	O
symplectic	B-Algorithm
integrators	I-Algorithm
is	O
formed	O
by	O
the	O
splitting	O
methods	O
.	O
</s>
<s>
Several	O
symplectic	B-Algorithm
integrators	I-Algorithm
are	O
given	O
below	O
.	O
</s>
<s>
A	O
third-order	O
symplectic	B-Algorithm
integrator	I-Algorithm
(	O
with	O
)	O
was	O
discovered	O
by	O
Ronald	O
Ruth	O
in	O
1983	O
.	O
</s>
<s>
Later	O
on	O
,	O
Blanes	O
and	O
Moan	O
further	O
developed	O
partitioned	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
for	O
the	O
integration	O
of	O
systems	O
with	O
separable	O
Hamiltonians	O
with	O
very	O
small	O
error	O
constants	O
.	O
</s>
<s>
The	O
new	O
Hamiltonian	O
is	O
advantageous	O
for	O
explicit	O
symplectic	B-Algorithm
integration	I-Algorithm
,	O
because	O
it	O
can	O
be	O
split	O
into	O
the	O
sum	O
of	O
three	O
sub-Hamiltonians	O
,	O
,	O
,	O
and	O
.	O
</s>
<s>
In	O
recent	O
decades	O
symplectic	B-Algorithm
integrator	I-Algorithm
in	O
plasma	O
physics	O
has	O
become	O
an	O
active	O
research	O
topic	O
,	O
because	O
straightforward	O
applications	O
of	O
the	O
standard	O
symplectic	O
methods	O
do	O
not	O
suit	O
the	O
need	O
of	O
large-scale	O
plasma	O
simulations	O
enabled	O
by	O
the	O
peta	O
-	O
to	O
exa-scale	O
computing	O
hardware	O
.	O
</s>
<s>
Special	O
symplectic	B-Algorithm
algorithms	I-Algorithm
need	O
to	O
be	O
customarily	O
designed	O
,	O
tapping	O
into	O
the	O
special	O
structures	O
of	O
physics	O
problem	O
under	O
investigation	O
.	O
</s>
<s>
To	O
overcome	O
this	O
difficulty	O
,	O
we	O
can	O
explore	O
the	O
specific	O
way	O
that	O
the	O
-dependence	O
and	O
-dependence	O
are	O
entangled	O
in	O
this	O
Hamiltonian	O
,	O
and	O
try	O
to	O
design	O
a	O
symplectic	B-Algorithm
algorithm	I-Algorithm
just	O
for	O
this	O
or	O
this	O
type	O
of	O
problem	O
.	O
</s>
<s>
First	O
,	O
we	O
note	O
that	O
the	O
-dependence	O
is	O
quadratic	O
,	O
therefore	O
the	O
first	O
order	O
symplectic	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
implicit	O
in	O
is	O
actually	O
explicit	O
.	O
</s>
<s>
This	O
is	O
what	O
is	O
used	O
in	O
the	O
canonical	O
symplectic	O
particle-in-cell	B-Algorithm
(	O
PIC	O
)	O
algorithm	O
.	O
</s>
<s>
To	O
build	O
high	O
order	O
explicit	O
methods	O
,	O
we	O
further	O
note	O
that	O
the	O
-dependence	O
and	O
-dependence	O
in	O
this	O
are	O
product-separable	O
,	O
2nd	O
and	O
3rd	O
order	O
explicit	O
symplectic	B-Algorithm
algorithms	I-Algorithm
can	O
be	O
constructed	O
using	O
generating	O
functions	O
,	O
and	O
arbitrarily	O
high-order	O
explicit	O
symplectic	B-Algorithm
integrators	I-Algorithm
for	O
time-dependent	O
electromagnetic	O
fields	O
can	O
also	O
be	O
constructed	O
using	O
Runge-Kutta	B-Algorithm
techniques	O
.	O
</s>
<s>
General	O
symplectic	B-Algorithm
integrator	I-Algorithm
for	O
non-constant	O
non-canonical	O
symplectic	O
structure	O
,	O
explicit	O
or	O
implicit	O
,	O
is	O
not	O
known	O
to	O
exist	O
.	O
</s>
<s>
However	O
,	O
for	O
this	O
specific	O
problem	O
,	O
a	O
family	O
of	O
high-order	O
explicit	O
non-canonical	O
symplectic	B-Algorithm
integrators	I-Algorithm
can	O
be	O
constructed	O
using	O
the	O
He	O
splitting	O
method	O
.	O
</s>
<s>
Then	O
explicit	O
high-order	O
non-canonical	O
symplectic	B-Algorithm
algorithms	I-Algorithm
can	O
be	O
constructed	O
using	O
different	O
compositions	O
.	O
</s>
<s>
A	O
-th	O
order	O
scheme	O
can	O
be	O
constructed	O
from	O
a	O
-th	O
order	O
scheme	O
using	O
the	O
method	O
of	O
triple	O
jump	O
,	O
The	O
He	O
splitting	O
method	O
is	O
one	O
of	O
key	O
techniques	O
used	O
in	O
the	O
structure-preserving	O
geometric	O
particle-in-cell	B-Algorithm
(	O
PIC	O
)	O
algorithms	O
.	O
</s>
