<s>
In	O
the	O
mathematical	O
field	O
of	O
numerical	B-Algorithm
ordinary	I-Algorithm
differential	I-Algorithm
equations	I-Algorithm
,	O
a	O
geometric	B-Algorithm
integrator	I-Algorithm
is	O
a	O
numerical	O
method	O
that	O
preserves	O
geometric	O
properties	O
of	O
the	O
exact	O
flow	O
of	O
a	O
differential	O
equation	O
.	O
</s>
<s>
We	O
can	O
motivate	O
the	O
study	O
of	O
geometric	B-Algorithm
integrators	I-Algorithm
by	O
considering	O
the	O
motion	O
of	O
a	O
pendulum	O
.	O
</s>
<s>
(	O
explicit	B-Algorithm
Euler	I-Algorithm
)	O
,	O
</s>
<s>
(	O
implicit	B-Algorithm
Euler	I-Algorithm
)	O
,	O
</s>
<s>
(	O
symplectic	B-Algorithm
Euler	I-Algorithm
)	O
,	O
</s>
<s>
(	O
Note	O
that	O
the	O
symplectic	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
treats	O
q	O
by	O
the	O
explicit	O
and	O
by	O
the	O
implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
)	O
</s>
<s>
trajectories	O
of	O
the	O
system	O
:	O
they	O
are	O
the	O
level	B-Algorithm
curves	I-Algorithm
of	O
.	O
</s>
<s>
and	O
implicit	B-Algorithm
Euler	I-Algorithm
methods	I-Algorithm
we	O
take	O
,	O
and	O
z0	O
=(	O
0.5	O
,	O
0	O
)	O
and	O
(	O
1.5	O
,	O
0	O
)	O
respectively	O
;	O
for	O
the	O
other	O
two	O
methods	O
we	O
take	O
,	O
and	O
z0	O
=(	O
0	O
,	O
0.7	O
)	O
,	O
(	O
0	O
,	O
1.4	O
)	O
and	O
(	O
0	O
,	O
2.1	O
)	O
.	O
</s>
<s>
implicit	O
)	O
Euler	B-Algorithm
method	I-Algorithm
spirals	O
out	O
from	O
(	O
resp	O
.	O
</s>
<s>
The	O
other	O
two	O
methods	O
show	O
the	O
correct	O
qualitative	O
behaviour	O
,	O
with	O
the	O
implicit	O
midpoint	O
rule	O
agreeing	O
with	O
the	O
exact	O
solution	O
to	O
a	O
greater	O
degree	O
than	O
the	O
symplectic	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
example	O
we	O
see	O
that	O
the	O
numerical	O
flow	O
of	O
the	O
explicit	B-Algorithm
Euler	I-Algorithm
method	O
is	O
not	O
area-preserving	O
;	O
viz.	O
,	O
</s>
<s>
A	O
similar	O
calculation	O
can	O
be	O
carried	O
out	O
for	O
the	O
implicit	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
</s>
<s>
However	O
,	O
the	O
symplectic	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
is	O
area-preserving	O
:	O
</s>
<s>
just	O
as	O
the	O
exact	O
flow	O
is	O
;	O
they	O
are	O
two	O
examples	O
of	O
geometric	O
(	O
in	O
fact	O
,	O
symplectic	B-Algorithm
)	O
integrators	O
.	O
</s>
<s>
Existing	O
methods	O
such	O
as	O
Runge-Kutta	B-Algorithm
can	O
be	O
modified	O
using	O
moving	O
frame	O
method	O
to	O
produce	O
invariant	O
versions	O
.	O
</s>
