<s>
In	O
Hamiltonian	O
mechanics	O
,	O
a	O
canonical	B-Algorithm
transformation	I-Algorithm
is	O
a	O
change	O
of	O
canonical	O
coordinates	O
that	O
preserves	O
the	O
form	O
of	O
Hamilton	O
's	O
equations	O
.	O
</s>
<s>
Canonical	B-Algorithm
transformations	I-Algorithm
are	O
useful	O
in	O
their	O
own	O
right	O
,	O
and	O
also	O
form	O
the	O
basis	O
for	O
the	O
Hamilton	O
–	O
Jacobi	O
equations	O
(	O
a	O
useful	O
method	O
for	O
calculating	O
conserved	O
quantities	O
)	O
and	O
Liouville	O
's	O
theorem	O
(	O
itself	O
the	O
basis	O
for	O
classical	O
statistical	O
mechanics	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
coordinate	O
transformations	O
(	O
also	O
called	O
point	B-Algorithm
transformations	I-Algorithm
)	O
are	O
a	O
type	O
of	O
canonical	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
class	O
of	O
canonical	B-Algorithm
transformations	I-Algorithm
is	O
much	O
broader	O
,	O
since	O
the	O
old	O
generalized	O
coordinates	O
,	O
momenta	O
and	O
even	O
time	O
may	O
be	O
combined	O
to	O
form	O
the	O
new	O
generalized	O
coordinates	O
and	O
momenta	O
.	O
</s>
<s>
Canonical	B-Algorithm
transformations	I-Algorithm
that	O
do	O
not	O
include	O
the	O
time	O
explicitly	O
are	O
called	O
restricted	O
canonical	B-Algorithm
transformations	I-Algorithm
(	O
many	O
textbooks	O
consider	O
only	O
this	O
type	O
)	O
.	O
</s>
<s>
(	O
Canonical	B-Algorithm
transformations	I-Algorithm
are	O
a	O
special	O
case	O
of	O
a	O
symplectomorphism	O
.	O
)	O
</s>
<s>
where	O
is	O
a	O
new	O
Hamiltonian	O
(	O
sometimes	O
called	O
the	O
Kamiltonian	B-Algorithm
)	O
that	O
must	O
be	O
determined	O
.	O
</s>
<s>
The	O
indirect	O
conditions	O
allow	O
us	O
to	O
prove	O
Liouville	O
's	O
theorem	O
,	O
which	O
states	O
that	O
the	O
volume	O
in	O
phase	O
space	O
is	O
conserved	O
under	O
canonical	B-Algorithm
transformations	I-Algorithm
,	O
i.e.	O
,	O
</s>
<s>
In	O
general	O
,	O
the	O
scaling	O
factor	O
is	O
set	O
equal	O
to	O
one	O
;	O
canonical	B-Algorithm
transformations	I-Algorithm
for	O
which	O
are	O
called	O
extended	O
canonical	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
Motion	O
itself	O
(	O
or	O
,	O
equivalently	O
,	O
a	O
shift	O
in	O
the	O
time	O
origin	O
)	O
is	O
a	O
canonical	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
translation	O
where	O
are	O
two	O
constant	O
vectors	O
is	O
a	O
canonical	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
the	O
Jacobian	O
matrix	B-Architecture
is	O
the	O
identity	O
,	O
which	O
is	O
symplectic	O
:	O
.	O
</s>
<s>
Set	O
and	O
,	O
the	O
transformation	O
where	O
is	O
a	O
rotation	O
matrix	B-Architecture
of	O
order	O
2	O
is	O
canonical	O
.	O
</s>
<s>
Be	O
aware	O
that	O
this	O
example	O
only	O
works	O
in	O
dimension	O
2	O
:	O
is	O
the	O
only	O
special	O
orthogonal	O
group	O
in	O
which	O
every	O
matrix	B-Architecture
is	O
symplectic	O
.	O
</s>
<s>
Jacobian	O
matrix	B-Architecture
is	O
indeed	O
given	O
by	O
which	O
is	O
symplectic	O
.	O
</s>
<s>
The	O
change	O
of	O
variable	O
between	O
one	O
set	O
of	O
canonical	O
coordinates	O
and	O
another	O
is	O
a	O
canonical	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
The	O
first	O
major	O
application	O
of	O
the	O
canonical	B-Algorithm
transformation	I-Algorithm
was	O
in	O
1846	O
,	O
by	O
Charles	O
Delaunay	O
,	O
in	O
the	O
study	O
of	O
the	O
Earth-Moon-Sun	O
system	O
.	O
</s>
