0 h8 y2d ff3 fs6 fc0 sc0 ls0 ws5">physical situation. By contrast, the canonical transforma-
tions locally map a 2n-dimensional space, the phase
space, into itself; but, the physical system under study
has n degrees of freedom. As we will see in a moment,
the n extra dimensions added to the configuration space
have the marvelous virtue of establishing a connection
between two given totally different mechanical systems
in the Hamiltonian formulation of classical mechanics.
The aim of the present work is to show, via the Ham-
ilton-Jacobi theory, that locally all the Hamiltonian sys-
tems with n degrees of freedom are equivalent to one
another. That is, there is a canonical transformation con-
necting two arbitrary Hamiltonian systems with the same
number of degrees of freedom. This in particular implies
that locally all the Hamiltonian systems are equivalent to
the free particle one. We illustrate our result with two
particular examples: we show that the one-dimensional
free particle is equivalent to the one-dimensional har-
monic oscillator and that the two-dimensional free parti-
cle is equivalent to the two-di mensional Kepler problem.
Several additional examples of this result have been pre-
sented in the literature [4-14]; but until now nobody has
presented a systematic procedure to compute the canoni-
cal transformation connecting the two given Hamiltonian
systems as we do here.
The organization of the present work is as follows: in
Section 2 we introduce the definition of a canonical
transformation and we remark that the canonical trans-
formation generated via a complete integral of the Ham-
ilton-Jacobi equation allows to stop the dynamics of an
arbitrary Hamiltonian system; this observation allow us
to establish our result in Section 3. In Section 4 we pre-
sent two particular examples and finally in Section 5 we
present our fin al discussion.
2. Canonical Transformations
Let (i
q,i) be local coordinates of the phase space asso-
ciated with a mechanical system with n degrees of free-
dom. The transformation
p

=,,
=,,
iijj
iijj
QQqpt
PPqpt
,
,
(1)
is canonical if and only if [3]

,= 0,,= 0,,=,
ijijij ij
QQPPQP
(2)
that is, if the Poisson brackets remain invariant under
such a transformation. A canonical transformation has
the remarkable property that a Hamiltonian system is
transformed into a new Hamiltonian system. The relation-
ship between the variables (i,i,q p
H
,t) characterizing
the original Hamiltonian system and the new variables
(i,i,, ) characterizing the new Hamiltonian
system, under a canonical transformation, is given by
Q p Kt
dd d=d
ii ii
pqQPH KtS , (3)
where
,,
jj
Sqpt is a generating function of the
canonical transformation. The canonical transformation
E. GALINDO-LINARES ET AL.
248
is explicitly obtained from
=
ii
S
pq
, (4)
=
ii
S
QP
. (5)
Furthermore,

=,,
ii S
KHqpt t
.
(6)
The Hamilton-Jacobi equation is obtained in the fol-
lowing way: first by using Equation (4) one replaces the
’s in Equation (6) and then one imposes the condition
. That is, the HJ equation is given by
i
p
K=0
,, =0. (7)
ii
SS
Hq t
qt





If a Hamiltonian system with Hamiltonian function,
,,
ii
H
qpt
,,
ii
SqPt , is given then a complete integral,
, of the Hamilton-Jacobi equation is the ge-
nerating function of a canonical transformation such that
the new Hamilton equations are given by
==0,
==
ii
ii
K
QP
K
PQ
0
,
,
(8)
That is, all the coordinates and momenta are constants
of motion. What does it mean? It means that the
extra dimensions added to the configuration space,
allows to stop the dynamics of the physical system under
study. In other words, the extra dimensions have the
virtue of making ignorable all the new coordinates of the
phase space (i
Q,i). In some sense, we can interpret this
important result saying that the Hamilton-Jacobi equation
provides a very special coordinate system, which is
“moving with the state of the physical system” and thus
there is no dynamics. We believe it is an intrinsic
property of the phase space associated with a physical
system with degrees of freedom. To clarify this point,
we assume we have obtained the solution to the original
Hamilton equations; that is, we have obtained
n
n
P
n


00
00
=,,
=,,
iijj
iijj
qqqpt
ppqpt
(9)
where and 0i are the values of i and i at
time respectively; that is, they are the initial
conditions. Then the point (i,i) of the phase space at
time , under the canonical transformation generated by
, is mapped to a new point with coordinates
(i
Q,i). This means that all the points of the phase space
described by Equation (9), with , are mapped to
the same point (,i). This is so because and
for
0i
q
=0
,,
jj
pt
p q
t
p
t
t
P
q p
0
Sq
1
t
i
Q P=0
i
Q
=0
i
P
1
0,t with 1. Therefore, we arrive
to the conclusion that a general canonical transformation
connecting two different Hamiltonian systems maps a
solution of the first Hamiltonian system to a solution of
the second Hamiltonian system. But the canonical
transformation generated by a complete integral of the
Hamilton-Jacobi equation maps a solution of the original
Hamiltonian system to a single point. In other words, the
canonical transformation, obtained via a complete inte-
gral of the HJ equation has the property of codifying the
entire information of the solution of the original Hamil-
tonian system into a single point.
t>0t
3. The Result
Theorem: Two arbitrary Hamiltonian systems with
degrees of freedom are locally equivalent. That is, there
is a canonical transformation connecting the two sets of
Hamilton equations.
n
systems with the same number of degrees of freedom
whose associated hamiltonian functions are denoted by
,,
jj
H
qpt and
,,
ii
qp

t
=,
=,
iij
j
q
qp
=,
=,
iij
j
q
qp


respectively. The Hami-
lton-Jacobi theory provides a canonical transformation
for each system


,
,
j
ii j
QQ pt
PP t
,
,
,
,
(10)
and


,
,
j
ii j
QQ pt
PP t

 (11)
respectively, such that both Hamiltonian systems are
equivalent to one with Hamiltonian function identically
zero. The desired canonical transformation relating the
original systems is obtained by requiring

=
=
i
ij
Qq
Pq


,,,
,,.
j jj
j j
pt pt
pt

,,
,,pt
ij
ij
Qq
q
P
P
Q
(12)
This result in particular implies that all the Hamiltonian
systems are locally equivalent to the free particle one.
Essentially what is happening is th at the first canonical
transformation, that given by Equation (10), takes a
solution of the first Hamiltonian system and maps it into
a single point with coordinates (i,i). The second
canonical transformation, that given by Equation (11),
takes a solution of the second Hamiltonian system and
maps it, in general, into a new single point with
coordinates (i,i). The condition given by Equation
(12) is saying that the first canonical transformation
codifies the entire information of a solution of the first
Q P
E. GALINDO-LINARES ET AL. 249
Hamiltonian system into a single point and that the
inverse of the second canonical transformation unfold
that information by using the second Hamiltonian and
thus a solution to the second Hamiltonian system is
obtained. Observe that a more general condition is that
given by


=,
=,
iijj
iijj
QQP
PQP
,
.
(13)
But, in the examples presen ted in this work, to obtain th e
canonical transformation connecting the two Hamiltonian
systems, we use the condition (12).
4. Examples
We illustrate the result stated above with two particular
examples, in the first one we show that the one-dimen-
sional free particle is equivalent to the one-dimensional
harmonic oscillator, in the second one we show that the
two-dimensional free particle is equivalent to the two-
dimensional Kepler problem.
4.1. The Free Particle and the Harmonic
Oscillator
The Hamiltonian function for the one-dimensional free
particle is gi ven by

2
,, =.
2
p
Hxpt m (14)
Therefore, the associ at ed HJ equation is gi v e n by
2
1=0.
2
SS
mx t




 (15)
A direct computation shows that a complete integral to
this equation is given by

,=2 ,SxPmPx Pt (16)
where P is a separation constant, which is identified with
the new momentum and corresponds to the energy, E, of
the free particle. Thus, the canonical transformation gen-
erated by this generating function is explicitly given by

,, =,
mx
Qxpt t
p (17)

2
,, =.
2
p
Pxpt m (18)
On the other hand, the Hamiltonian function describ-
ing the evolution of the one-dimensional harmonic oscil-
lator is given by

2222
,, =,
2
pmx
Hxpt m
%
%%% %
(19)
then the associated HJ equation is given by
2
222
1=0,
2
S
mx
mx t










%%
%
%
S
(20)
and a complete integral to this equ ation is

22
,,=2d1,
2
mx
SxptmP xPt
P

%
%%%
%%% % (21)
where is a separation constant, which is identified
with the new momentum and corresponds to the energy,
, of the physical system. For this case, the canonical
transformation generated by this generating function is
explicitly given by
P
%
E
%

2222
1
,, =arcsin,
mx
Qxpt t
pmx




%
%%%
%%
(22)

2222
,, =.
2
pmx
Pxpt m
%
%%% %
(23)
For this case, the condition (12) provides the desired
canonical transformation


2222
2222
2222
,, =arcsin,
,, =.
pmx mx
xxptmpmx
pxptp mx




%% %
%%
%%
%% %%
(24)
Or equivalentl y


,, =sin,
,, = cos.
pmx
xxptmp
mx
pxpt pp





%
%
(25)
A direct comput ati on sh o ws t hat
 
2222 2
,, ===,,.
22
pmx p
H
xptH xpt
mm
%%
%%% (26)
Observe that in the limit
going to zero the canoni-
cal transformation (24) reduces to the identity one. This
result is consistent with the fact that in this limit the two
Hamiltonians coinci de. Equation (25) are equ ivalent t o
i=expi
mx
pmxp p






%% .
(27)
Observe that this transformation is not one to one.
4.2. The Free Particle and the Kepler Problem
The Hamiltonian describing the evolution of a two-dim-
E. GALINDO-LINARES ET AL.
250
ensional free particle, in polar coordinates, is given by
2
2
2
1
=
2r
p
Hp
mr


.
(28)
Therefore, the HJ equation for this problem can be
written in the following form
22
2
11 =0.
2
SSS
mr t
r


 


 

 


(29)
Looking for a separable solution one finds that a com-
plete integral is given by

22
12
122 1
2
,,,,=d,
mr PP
SrtPPr PPt
r

(30)
where 1 and 2 are two constants of separation,
which are identified with the new momenta. A direct
computation shows that for this case the canonical trans-
formation is explicitly given by
P P
11
221
22 2
12
2
=,
2
=arcsin ,
2
=,
2
=.
r
r
rp
Qt
P
p
Qmr P
rp p
Pmr
Pp




(31)
On the other hand, the Hamiltonian describing the
evolution of the two-dimensional Kepler problem, by
using polar coordinates can be written in the following
form
2
22
1
=
2r
p
Hp
mr
r





,
(32)
where 12
and =Gmm
12 12
=mmmmm. Therefore,
the HJ equation for the Kepler problem is given by
22
2
11 =0.
2
SS S
mr rt
r

 
 


 

 

 

(33)
As in the previous cases, a direct computation shows
that a complete integral of this equation is given by

22
12
12
21
22
Any Hamiltonian System Is Locally Equivalent to a Free Particle