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
Copyright © 2012 SciRes. WJM
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
Proof: We start with two given different Hamiltonian
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
Copyright © 2012 SciRes. WJM
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-
Copyright © 2012 SciRes. WJM
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
World Journal of Mechanics, 2012, 2, 246-252
doi:10.4236/wjm.2012.25030 Published Online October 2012 (http://www.SciRP.org/journal/wjm)
Any Hamiltonian System Is Locally Equivalent to a Free
Particle
Elizabeth Galindo-Linares1, Esperanza Navarro-Morales1, Gilberto Silva-Ortigoza1*,
Román Suárez-Xique1, Magdalena Marciano-Melchor2, Ramón Silva-Ortigoza2,
Edwin Román-Hernández3
1Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, México
2CIDETEC-IPN, Departamento de Posgrado, Área de Mecatrónica, Unidad Profesional Adolfo López Mateos,
México City, México
3Departamento de Matemáticas Aplicadas, Universidad del Istmo, Guatemala, México
Email: *gsilva@fcfm.buap.mx
Received July 19, 2012; revised August 16, 2012; accepted August 26, 2012
ABSTRACT
In this work we use the Hamilton-Jacobi theo ry to show that locally all the Hamiltonian systems with n degrees of free-
dom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltonian systems with the
same number of degrees of freedom. This result in particular implies that locally all the Hamiltonian systems are
equivalent to that of a free particle. We illustrate our result with two particular examples; first we show that the
one-dimensional free particle is locally equivalent to the one-dimensional harmonic oscillator and second that the
two-dimensional free particle is locally equivalent to the two-dimensional Kepler problem.
Keywords: Hamiltonian System; Canonical Transformation
1. Introduction
In the Lagrangian formulation of classical mechanics to
each mechanical system with n degrees of freedom there
is associated an abstract space of dimension n, the con-
figuration space, and a scalar function, the Lagrangian,
which encodes the nature of the dynamics of the me-
chanical system [1-3]. The state of the physical system at
a given time is represented by a point in the configura-
tion space and its evolution between two given states,
two points in the configuration space, is given by the
Hamilton principle, which establishes that to the motion
of the physical system between two given states there is
associated a curve on the configuration space such that
the first variation of the action is zero. That is, among all
the possible curves in the configuration space connecting
the two points associated with the two given states, the
evolution of the physical system singles out that curve
obtained from the condition that the first variation of the
action is zero. For mechanical systems with holonomic
constrains, the Hamilton principle is both a necessary and
sufficient condition for Lagrange’s equations. Since the
Lagrangian we are interested in is a function of the gen-
eralized coordinates, the generalized velocities and the
time, then the Lagrange equations describing the evolu-
tion of a mechanical system with n degrees of freedom is
a set, in general coupled, of second-order ordinary dif-
ferential equations for the generalized coordinates. The
complexity in solving this system of equations is due to
two different reasons: a bad election in the used coordi-
nates and the nature of the force. To integrate the La-
grange equations we have to our disposition the remark-
able property of invariance of these equations under a
point transformation. Using this property on e look s for, if
necessary, a set of point transformations such that in the
final Lagrangian one has as many ignorable coordinates
as possible, this way the associated generalized momenta
are conserved and the final equations are easier to inte-
grate than the original ones. It is important to emphasize
that the original Lagrange equations and those obtained
under a point transformation describe exactly the same
mechanical system. That is, in the Lagrangian formula-
tion of classical mechanics two different mechanical
systems with the same number of degrees of freedom can
not be conn ected via a point tran sformation.
In the Hamiltonian formulation of classical mechanics
to each mechanical system with n degrees of freedom
there is associated another kind of abstract space of di-
mension 2n, the phase space, a symplectic structure on it,
the Poisson brackets, and a scalar function, the Hamilto-
nian, which encodes the nature of the dynamics of the
physical system. In this formulation of classical mechan-
*Corresponding a uthor.
Copyright © 2012 SciRes. WJM
E. GALINDO-LINARES ET AL. 247
ics the generalized coordinates and momenta have the
same status, the two sets form a local coordinate system
of the phase space. This means that from a mathematical
point of view there is no difference between coordinates
and momenta. This is what is remarked as one of the
astonishing properties of the Hamiltonian formulation.
But, what are the physical consequences, if any, derived
from this result? This in particular means that there is no
difference between kinetic and poten tial energies; that is,
they are at the same status too, what really matters in this
formulation is the total energy or more generally the
Hamiltonian. In particular, this result implies that for any
Hamiltonian system there must exist a local transforma-
tion that allows to “transfer” the potential energy into the
kinetic energy and thus be equivalent to a free particle. In
this formulation the dynamics is given by the modified
Hamilton principle, which establishes that the motion of
the physical system characterized by the coordinates and
momenta at a given time, the initial conditions, singles
out in the phase space a curve obtained from the condi-
tion that the first variation of the action written in terms
of coordinates, momenta and their time derivatives is
zero. In this case, the dynamics of the physical system is
given by the Hamilton equations, which for a system
with n degrees of freedom is a system, in general coupled,
of 2n first-order ordinary differential equations for the
coordinates and momenta. Since the points of the phase
space are labeled by the coordinates and momenta, the
analog of the point transformations in the Hamiltonian
formulation is a set of transformations that in general mix
the original coordinates and momenta. There is a special
class of transformations in the phase space, they are the
canonical transformations, singled out by requiring the
invariance of the Poisson brackets. This in turn, implies
that the Hamilton equations be invariant in form under a
canonical transformation. The point transformations of
the configuration space are a very particular subset of
these more general transformations. The property of in-
variance of the Hamilton equations under a canonical
transformation has been used to integrate the Hamilton
equations, here one normally looks for a canonical trans-
formation such that the new Hamilton equations be sim-
pler that the original ones.
It is important to remark that for a Lagrangian system
with n degrees of freedom the point transformations lo-
cally map an n-dimensional space, the configuration
space, into itself. Therefore, as we have mentioned be-
fore, the original Lagrange equations and those obtained
under a point transformation describe exactly the same
,,, ,=d
,
mr PmrP
Sr tPPr
r
PPt







(34)
where 1 and 2 are two constants of separation.
Another direct computation shows that the canonical
transformation is explicitly given by
P
P

1
111
11
2
222 222
1
22 2
12
2
=222
log 222,
=arcsin ,
2
=,
2
=.
r
r
r
rp m
QPPP
mmrP rpmPt
pmr
QmrP pmr
rp p
Pr
mr
Pp








%
%
%
%
%
%
%
%%
%
%%%
%%
%%%
%%
%%
%
%% %
%%%
%
%
%
%%
   
(35)
Observe that for =0
, the Hamiltonian describing
the Kepler problem reduces to the free particle case. As
should be, in this particular case, the above transforma-
tion reduces to that obtained for the free particle case.
From Equations (31) and (35), and condition (12), we
finally obtain that the transformation connecting the
two-dimensional free particle with the two-dimensional
Kepler probl em is give n by

22
22 2
2
22 222
1
21
=,
2
=arcsin
2
arcsin ,
2
=,
=,
r
r
r
r
rp gp
rr
rp pmr
pmr
mrP pmr
p
mr P
rp g
pr
pp











%
%
%
%
%
%
%
%
%
%% %
%
%% %%
%%
%
%
%% %
%
%
%%
%
(36)
where

2
1
1
log 222.
2r
m
gmmrPrp
P
1
mP

%
%%
%%%
% (37)
5. Discussion
We know that the configuration space associated with a
Lagrangian system with degrees of freedom is a d iff-
erentiable manifold of dimension ; this means that it is
locally equivalent to an open subset of [3]. Therefore,
a solution to the Lagrange equations, a curve on the
configuration space, is locally equivalent to a straight
line. However, this does not mean that via a point
transformat ion one can connect two arbitrary Lagran gian
systems with the same number of degrees of freedom.
For example, the one-dimensional free particle cannot be
transformed, via a point transformation, into the one-
dimensional harmonic oscillator. On the other hand, the
phase space associated with a Hamiltonian system with
nnn
R
Copyright © 2012 SciRes. WJM
E. GALINDO-LINARES ET AL. 251
n degrees of freedom is another differentiable manifold,
a symplectic manifold, of dimension and thus it is
locally equivalent to an open subset of [3]. The
canonical transformations, which are defined as those
transformations of the phase space such that the Poisson
brackets or symplectic structure remains invariant have
the property of connectin g two Hamiltonian syste ms with
the same number of degrees of freedom. This result is
well established and well known. But one of the main
contributions of this work was to realize that the very
special class of canonical transformations generated via a
complete integral of the HJ equation has the property of
mapping an entire curve in to a single point. This property
was used to show that all the Hamiltonian systems with
the same number of degrees of freedom are locally
equivalent. Even though all the Hamiltonian systems are
locally equivalent to each other, at the Lagrangian level
they, in general, give rise to physically nonequivalent
Lagrangian systems. For example, as we have shown, the
one-dimensional free particle and the one-dimensional
harmonic oscillator are locally equivalent in the Hami-
ltonian formulation in the sense that there is a canonical
transformation that maps one system into the other. But,
at the Lagrangian level they cannot be connected via a
point transformation; that is, they are two Lagrangian
systems describing two different physical situations.
Observe that in the Lagrangian formulation the property
of invariance in form of Lagrange equations and the fact
that the dimension of the configuration space is equal to
the number of degrees of freedom of the system, implies
that the original Lagrangian system and that obtained
under a point transformation describe exactly the same
mechanical system. In some sense, in Lagrangian formu-
lation, mathematical equivalence implies physical equi-
valence. However, in the Hamiltonian formulation it is
not true, and it is so, because the dimension of the phase
space is two times the number of degrees of freedom of
the physical system. As we have remarked, for each
Hamiltonian system the extra dimensions added to
the configuration space allows to find a co ord ina te sys tem
where there is no dynamics. This very important property
cannot be found in the Lagrangian formulation.
2n2n
R
n
We finish our work with the following observation: we
start with a physical system such that its d ynamics in the
Lagrangian formulation is described by the lagrangian
function, . Furthermore, we assume that its associated
natural Hamiltonian,
L
H
, can be computed (the natural
Hamiltonian is that obtained via a Legendre transfor-
mation). Therefore, at this stage, we have a Hamiltonian
system with Hamiltonian function,
H
. Now by perfo-
rming a canonical transformation on it we obtain a new
Hamiltonian system with Hamiltonian function,
H
. If
the natural Lagrangian function, , associated with the
new Hamiltonian system is defined, then at the end we
have two Lagrangian systems, our observation is that the
new Lagrangian system may not describe the original
physical system. If the canonical transformation used to
obtain the new Hamiltonian system corresponds to a
point transformation then both Lagrangian systems des-
cribe the same physical situation, but if the used ca-
nonical transformation does not come from a point trans-
formation, then the two Lagrangian systems describe two
different physical situations. We emphasize that there are
canonical transformations that give rise to Hamiltonian
systems such that its natural Lagrangian is not defined.
L
6. Acknowledgements
E. Galindo-Linares and E. Navarro-Morales were sup-
ported by a CONACyT scholarship. G. Silva-Ortigoza
acknowledges financial support from SNI (México). This
work has received partial support from VIEP-BUAP. R.
Silva-Ortigoza and M. Marciano-Melchor acknowledge
financial support from SNI, Secretar de Investigación y
Posgrado del IPN (SIP-IPN) and the programs EDI, EDD
and COFAA of IPN.
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