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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

H

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