Embed Size (px)
Citation preview
A. La Rosa Lecture Notes PSU-Physics PH 411/511 ECE 598
I N T R O D U C T I O N T O Q U A N T U M M E C H A N I C S
________________________________________________________________________ From the Hamilton’s Variational Principle to the Hamilton Jacobi Equation
4.1. The Lagrange formulation and the Hamilton’s variational principle 4.1A Specification of the state of motion
4.1B Time evolution of a classical state: Hamilton’s variational principle Definition of the classical action The variational principle leads to the Newton’s Law The Lagrange equation of motion obtained from the variational principle
Example: The Lagrangian for a particle in an electromagnetic field 4.1C Constants of motion Cyclic coordinates and the conservation of the generalized momentum Lagrangian independent of time and the conservation of the Hamiltonian Case of a potential independent of the velocities: Hamiltonian is the
mechanical energy
4.2 The Hamilton formulation of mechanics 4.2A Legendre transformation
4.2B The Hamilton Equations of Motion Recipe for solving problems in mechanics Properties of the Hamiltonian 4.2C Finding constant of motion before calculating the motion itself Looking for functions whose Poisson bracket with the Hamiltonian vanishes Cyclic coordinates
4.2D The modified Hamilton’s principle: Derivation of the Hamilton’s equations from a variational principle.
4.3 The Poisson bracket 4.3A Hamiltonian equations in terms of the Poisson brackets 4.3B Fundamental brackets 4.3C The Poisson bracket theorem: Preserving the description of the classical
motion in terms of a Hamiltonian a) Example of motion described by no Hamiltonian b) Change of coordinates to attain a Hamiltonian description
4.4 Canonical transformations 4.4A Canonoid transformations (i.e. not quite canonical)
Preservation of the canonical equations with respect to a particular Hamiltonian)
4.4B Canonical transformations Definition Canonical transformation theorem Canonical transformation and the invariance of the Poisson bracket
4.4C Restricted canonical transformation
4.5 How to generate (restricted) canonical transformations
4.5A Generating function of transformations
4.5B Classification of (restricted) canonical transformations
4.5C Time evolution of a mechanics state viewed as series of canonical transformations
The generator of the identity transformation Infinitesimal transformations The Hamiltonian as a generating function of canonical transformations Time evolution of a mechanical state viewed as a canonical transformation
4.6 Universality of the Lagrangian 4.6A Invariant of the Lagrangian equation with respect to the configuration space
coordinates 4.6B The Lagrangian equation as an invariant operator
4.7 The Hamilton Jacobi equation The Hamilton principal function Further physical significance of the Hamilton principal function
From the Hamilton’s Variational Principle to the Hamilton Jacobi Equation
4.1 The Lagrange formulation and the Hamilton’s variational principle
4.1A Classical specification of the state of motion
The spatial configuration of a system composed by N point masses is completely described by
3N Cartesian coordinates (x1, y1, z1), (x2, y2, z2), …, (xN , yN , zN ).
If the system is subjected to constrains then the 3N Cartesian coordinates are not
independent variables. If n is the least number of variables necessary to specify the most general motion of the system, then the system is said to have n degrees of
freedom.
The configuration of a system with n degrees of freedom is fully specified by n generalized position-coordinates q1, q2, … , qn
The objective in classical mechanics is to find the trajectories,
q = q (t) = 1 , 2, 3, …, n (1)
or simply q = q (t) where q stands collectively for the set (q1, q2, … qn)
4.1B Time evolution of a classical state: Hamilton’s variational principle One of the most elegant ways of expressing the condition that determines the
particular path )(tq that a classical system will actually follow, out of all other possible
paths, is the Hamilton’s principle of least action, which is described below.
The classical action
One first expresses the Lagrangian L of the system in terms of the generalized
position and the generalized velocities coordinates q and q ( =1,2,…, n).
L( q , q , t ) (2)
Typically L=T-V, where T is the kinetic energy and V is the potential energy of the system. For example, for a particle of mass m moving in a potential V(x,t), the Lagrangian is,
L (x, x , t) = (1/2)m x 2 - V(x,t) (3)
Then, for a couple of fixed end points (a,t1) and (b, t2) the classical action S is defined as,
S (q ) ≡
)(
)(
2
1
t b,
ta,L( )(tq , )(tq , t ) dt Classical action (4)
Function or set of functions (n-dimension case)
Number (1-dimension case) or set of numbers (n-dimension case)
Notice S varies depending on the arbitrarily selected path q that joins the ending point a and b, at the corresponding times t1 and t2. See figure 1 below.
We emphasize that in (4), q stands for a function (or set of functions, in the n-dimension case) Indistinctly we will also call it a path
q(t) is a number (or set of numbers, in the n-dimension case) It is the value of the function q when the argument is t.
S is evaluated at a function), i.e. S (q)
L is evaluated at a number (or set of numbers, in the n-dimension case)
Hamilton’s variational principle for conservative systems
Out of all the possible paths that go from (a, t1) to (b, t2), the system takes only one.
On what basis such a path is chosen?
Answer: The path followed by the system is the one that makes the functional S an extreme (i. e. a maximum or a minimum).
S qq = 0 The variational principle (5)
That is, out of all possible paths by which the system could travel from an initial position at time t1 to a final position at time t2, it will actually travel along the path for which the integral is an extremum, whether a maximum or a minimum. The figure below shows the case for the one-dimensional motion.
x(t)
t t1 t2
xC
a
b
xC(t)
Figure 1. For a fixed points (a, t1) and (b, t2), among all the possible paths with the same end points, the path xC makes the action S an extremum.
Example: The variational principle leads to the Newton’s law Consider a particle moving under the influence of a conservative force F (be
gravitational force, spring force, …) whose associate potential is V (i.e. F = -dx
dV.)
Since the kinetic energy is given by T= (1/2) m x 2, where x means dx/dt, then
L(x, x , t) = (1/2) m x 2 - V(x), and
S(x) ≡ )(
)()(
2
1
tb,
ta,dtt,xx,L =
)(
)( 2
12
1
][ )(tb,
ta,dtxVxm 2 (6)
On the left side, S is evaluated on a path x.
On the right side, the values x(t) and x (t) have to be used inside the integral, but for simplicity we have just written x
and x respectively
Different paths x give, in general, different values for S (for fixed (a, t1) and (b, t2.)
Let’s assume xC is the path that makes the value of S extremum. One way to obtain
a more explicit form of xC is to probe expression (6) with a family of trial paths that are infinitesimally neighbors to that particular path. Let’s try, for example,
x( = xC + h (7)
Scalar
(parameter)
Function
which means
x(( t) = xC(t) + h(t) (7)’
where h(t) is an arbitrary function subjected to the condition,
h(t1)= h(t2) =0 (8)
and is an arbitrary scalar parameter,
In expression (7), when choosing small values for , x( becomes a path just a
“bit” different than the path xC.
Note: In the literature the path difference [x( - xC ] is sometimes called x.
Here we are opting to use h ( a number, and h a function) instead, just
to emphasize that [x(-xC] is a difference between two paths (and not the difference of just two numbers).
Evaluating expression (6) at the path x(( gives,
S(x() = )(
)(
2
1
t b,
t a,{ m
2
1 [ x C(t) + h (t) ]2
– V( xC(t ) + h(t ) ) }dt (9)
The condition that xC is an extremum becomes,
00
d
dS (10)
x(t)
t t1 t2
h(t)
a
b xC(t)
Figure 2. For an arbitrary function h, a parametric family of trial paths x(t ) =
xC(t ) + h(t ) is used to probe the action S given by expression (6).
From (6),
dα
dS
)(
)(
2
1
t b,
t a,{ m
2
12 [ x C(t) + h (t)] h (t) –V ’h(t) }dt
where V’ stands for the derivative of the potential V
= )(
)(
2
1
t b,
t a,{ m x C (t) h (t) – V ’ h(t) }dt +
)(
)(
2
1
t b,
t a, m [ h (t)]2dt
The last term will be cancelled when evaluated at =0.
0d
dS
)(
)(
2
1
t b,
t a,{ m x C (t) h (t) – V ’(xC) h(t) }dt (11)
The first term on the right side of the equality above can be integrated by parts,
)(
)(
2
1
t b,
t a,x C (t) h (t) dt = x C (t)
2
1
)(t
tth -
)(
)(
2
1
t b,
t a,x C (t) h(t) dt
According to (8), h(t) vanishes at t1 and t2 ; therefore the first terms on the right side vanishes,
)(
)(
2
1
t b,
t a,x C (t) h (t) dt = -
)(
)(
2
1
t b,
t a,x C (t)h(t) dt
Expression (11) becomes,
0d
dS –
)(
)(
2
1
t b,
t a,{ m x C (t) + V ’(xC)} h(t)dt (12)
Since this last expression has to be equal to zero for any function h(t), it must happen
that m x C (t) + V ’(xC) =0, or,
x C (t) = – dx
dV(xC) = F
That is,
S ≡ )(
)(
2
1
t b,
t a,t)dt,xL(x, =
)(
)( 2
11
1
][ )(t b,
t a,dtxVxm 2 ….
takes an extreme value when the path x(t) . (13)
satisfies the Newton’s Law x (t) = – dx
dV = F
(13)
In the next section, a key expression we will use through the derivation is the following:
Consider an arbitrary function, )( t,,SS vu ,
where )u,u,u( 321u and )v,v,v( 321v
For arbitrary increments , , and the value of
)( dt t , , S δvΔu
can be approximated by,
)( )( t,,Sdtt , , S vuδvΔu +
+ 3
3
2
2
1
1 uuu
SSS +
+ 3
3
2
2
1
1 vvv
SSS
+ dtt
S
Sometimes the following notation is used,
)u
,u
,u
( 321
SSSS
u
Accordingly,
)( )( t,,Sdtt , , S vuδvΔu + v
δu
Δ
SS + dt
t
S
(13)
The Lagrange equations of motion obtained from the Hamilton’s variational principle In a more general case, the system may be composed by n particles. Using,
x (t)= (x1(t), x2(t), … , xn(t), 1x (t), 2x (t), ,…, nx (t) )
the action is defined as
S( x
) ≡ )(
)(
2
1
t ,
t ,
b
aL( x (t), x (t) , t) dt (14)
where a and b are two fixed point in the configuration space. Using a family of trial functions of the form
x ( (t) = x (t) + h (t) (15)
where h (t1) = h (t2) = 0
x (t)= ( x1(t), x2(t), … , xn(t), 1x (t), 2x (t), ,…, nx (t) )
h (t)= ( h1(t), h2(t), … , hn(t), 1h (t), 2h (t), ,…, nh (t) )
we look for an extreme value of the function
S() = 2
1
t
tL( x (
(t), x ((t), t) dt (16)
From (15) and (16) one obtains,
d
dS
2
1
t
t{ i
ii
i
ii
hx
Lh
x
L
} dt
Notice, the term t
L
does not appear in the last expression.
Integrating by parts the second term,
d
dS
2
1
t
t{ i
ii
hx
L
} dt +
2
1
t
t
i
ii
hx
L
-
2
1
t
t{ i
ii
hx
L
dt
d)(
} dt
Since h
(t1) = h
(t2) = 0, one obtains
d
dS
2
1
t
t{ i
ii
hx
L
} dt -
2
1
t
t{ i
ii
hx
L
dt
d)(
} dt
= i
2
1
t
t{
ix
L
- )(
ix
L
dt
d
} ih dt
In the last expression ix
L
and )(
ix
L
dt
d
are evaluated at x
(t) + )(th
.
0d
dS
i
2
1
t
t{
ix
L
- )(
ix
L
dt
d
} )(thi dt
In the last expression ix
L
and )(
ix
L
dt
d
are evaluated at x
(t).
The variational principle requires that,
0d
dS
i
2
1
t
t{
ix
L
- )(
ix
L
dt
d
} )(thi dt = 0 (17)
for arbitrary trial functions )(thi .
This could be satisfied only if,
0x
L
x
L
dt
d
ii
for i=1,2, …, n. The Lagrange Equations (18)
These are second order equations. The motion is completely specified if the initial
values of the n coordinates xi and the n velocities ix are specified. That is, the xi and ix
form a complete set of 2n independent variables for describing the motion.
Remark: Hamilton’s variational principle involves physical quantities T
and V, which can be defined without reference to a particular set of generalized coordinates. The set of Lagrange equations is therefore invariant with respect to the choice of coordinates. !
A key expression we will use through the derivation is the following:
For an arbitrary function, )( t,,SS vu ,
3
3
2
2
1
1
uu
uu
uu
SSS
dt
dS
+ 3
3
2
2
1
1
vv
vv
vv
SSS+
+ t
S
Sometimes, the following notation is used )u
,u
,u
( 321
SSSS
u
Accordingly,
dt
dS
vv
uu
SS +
t
S
Example: The Lagrangian for a particle in an electromagnetic field This is a case where the force depend on the velocity,
)v ( BEF q (19)
Since the Maxwell’s equation states that 0 B , then B can be expressed in terms of
a vector potential )( t,xAA
AB (20)
(because the divergence of the rotational is identically equal to zero.)
On the other hand, the third Maxwell’s equation 0/ tBE can then be written
as 0)(
t
AE , or 0)/( tAE . The quantity with vanishing curl can be
written as the gradient of some scalar function, that is t/AE , or
t
AE
(21)
where )( t,x
With these alternative expressions for E and B the equation of motion
)v ( BEx qm
can be expressed as,
t
qqm
Ax + )( Ax q (22)
It can be demonstrated (see homework assignment) that,
)( Ax )( -
Ax
x
Ax (23)
Accordingly, the vector expression (22) can be expressed in term of the components,
t
A
xx
q
m
+ )(
Ax
x
Ax = 1, 2, 3
The second and the fourth terms on the right side constitute a
exact differential of A
x
+
dt
dA
x
Ax (24)
We have to figure out the Lagrangian L, such that the Lagrangian equations (18) lead to
(24). If there were no magnetic field, L would be )( )( 2xx qm1/2L . The magnetic
field introduces a term depending on the velocity. Hence, let’s try,
)()( 2
2
1t,,fqmL xxxx (25)
where ),,( 322 xxxx
x
f
xq
x
L
x
fxm
x
L
x
f
tx
f
x
fxm
x
L
dt
d
)()(
xx
xx
where x
stands for ),,(
321 xxx
x
stands for ),,(
321 xxx
0
x
L
x
L
dt
d
implies
x
f
tx
f
x
f
x
f
xqxm
)()(
xx
xx (26)
which should be compared with (24)
xm x
q
+
dt
dAq
xq
Ax
Notice the right side of (26) contains a term in x , which is not present in (24). Therefore
we require
x
f
)(
xx = 0
Or, equivalently,
x
f
x
for = 1, 2, 3 (27)
A particular solution of (27) is,
321 GxGxGxt,f 321 )( xGx (28)
Replacing (28) in (26),
Gt
Gxx
qxm
)(
xx
Gx
The last two terms on the right side constitute a
exact differential of G
Gdt
d
xxqxm
Gx (29)
which should be compared with (24),
xm x
q
+
dt
dAq
xq
Ax
A comparison between (24) and (29) indicates that AG q . Replacing this solution in
(28) gives,
)( t,qf xAx (30)
The Lagrangian in (25) is then given by,
)( )( )( 2
2
1t,qqmt,,L xAxxxxx (31)
Lagrangian for a particle in an electro- magnetic field described by a scalar potential and vector potential A.
t
AE
AB
4.1C Constants of motion Cyclic coordinates and the conservation of the generalized momentum
There is another way to identify constant of motion. For example, if the Lagrangian
of the system does not contain a given coordinate x, the corresponding Lagrangian
equation 0
x
L
x
L
dt
d
becomes
0
x
L
dt
d
It means that the generalized momentum x
L
is constant
If a coordinate x does not appear in the Lagrangian, the variable is said to be cyclic.
p = constant for a cyclic coordinate (32)
Lagrangian independent of time and conservation of the Hamiltonian
Consider a Lagrangian that does not depend explicitly on time, i.e. t
L
= 0.
The total time derivative of
L L(x, x ) (33) is,
dt
dL
i
[ ix
L
dt
dxi +
ix
L
dt
xd i
]
From the Lagrange equation ii x
L
dt
d
x
L
, the previous equation can be written as
dt
dL
i
[ (ix
L
dt
d
)
dt
dxi +
ix
L
dt
xd i
]
i
[ (ix
L
dt
d
)
dt
dxi +
ix
L
dt
xd i
]
i
( )(i
ix
Lx
dt
d
dt
d
i
i
ix
Lx
This implies,
i
i
ix
Lx
L = constant (34)
when L does not depend on t explicitly.
We have just found a constant of motion.
The quantity on the left of expression (34) is called the Hamiltonian H of the system.
H ≡ i
i
ix
Lx
L (35)
A more rigorous definition of the Hamiltonian function will be given in the following sections.
Expression (34) indicates that, when L does not depend explicitly on time, the Hamiltonian quantity H is a constant of motion. Case of a potential independent of the velocities
For the case in which V = V(x),
L(x, x ) = T( x ) V(x)
= k
2
2
1kk xm V(x) (36)
ix
L
= ii xm
i
i
ix
Lx
=
i
2
ii xm
= 2 T (x)
H ≡ i
i
ix
Lx
L
= 2 T ( x ) L
= 2 T ( x ) ( T( x ) V(x) )
= T( x ) V(x) = Energy (37)
Thus, when the potential does not depend on the the velocity, and L does not depend on the time explicitly, H is the energy of the system and at the same time a constant of motion.
4.2 The Hamiltonian formulation of mechanics An alternative to the Lagrangian description of mechanics, outlined above, is the
Hamiltonia formulation. Instead of dealing with a set of n differential equations of 2nd-order given in (18), one is resorted to solve 2n differential equations of 1st order, as we will see below. However one may end up with a similar intensity of difficulty when solving the corresponding equations. The advantages of the Hamiltonian formulation lie not necessarily in its use as a calculation tool, but rather in the deeper insight it affords into the formal structure of mechanics.1 Its more abstract formulation is of interest because of their essential role in constructing the more modern theories in physics. In this course it is used as a point of departure for elaborating a quantum theory.
In this section we show that the new formulation is implemented through:
a change of variables (to be specified later),
(x1, x2, …, xn, 1x , 2x , …, nx , t) later specifiedbe to (x1, x2, … xn, p1, p2, … pn,, t)
and, instead of ),,( tL xx , the use another function
H=H(x1, x2, …, xn, p1, p2, …,pn, t) = H(x, p, t)
How to obtain such a transformation? How to choose H? To get an idea of what transformation is convenient (i. e. what particular combination between the ),,( txx and ),,( tpx variables is suitable for our purpose), let’s familiarize
with the following, much simpler, Legendre transformation
4.2A Legendre transformation Consider a physical quantity is described by a two-variable function
L=L(x, y) (32)
A differential change of L is given by,
dL =x
L
dx +
y
L
dy
= u dx + v dy (33)
It may happen that a description could more convenient in terms of x and y
L
.
Accordingly, we would like, then, to perform a change of variables
(x, y) (x, v) (34)
where v = y
y)(x,L
(We will assume that the relationship v = y
y)(x,L
allow us to put y in terms of x and v).
One way to obtain a quantity B (to replace L) whose differential-change could be expressed directly in terms of dx and dv (instead of dx and dy) is by defining B=B(x, v) as follows,
B(x, v) ≡ L(x, y) – y v (35)
= L(x, y) – y y
y)(x,L
Its corresponding differential is given by
dB = dL – y dv – v dy
using (33)
= u dx + v dy – y dv – v dy
dB = u dx – y dv (36)
(As we said above, we assumed that the definition v = y/yx, L )( allows to
put y in terms of x and v. Also, since u= x L/ , the above assumption ensures u can be put in terms of x and v as well.)
Thus, dB end ups being expressed in terms of the differentials dx and dv (the new independent variables,) which is what we wanted.
The last expression also gives,
u =x
B
and y = –
v
B (37)
In summary,
(x, y) (x, v)
(x, v) = (x, y/ L )
L(x, y) B(x, v)
B(x, v) ≡ L(x, y) – y v
= L(x, y) – y y/ L
dL = ( x L/ ) dx + ( y L/ )dy = u dx + v d y
dB = u dx – y dv
(38)
With this background, we will see below that the transformation to transit from the Lagrangian formulation to the Hamiltonian formulation is of the Legendre’s type. (Just identify the x variable with the y variable we have used in this section.)
4.2B The Hamilton Equations of Motion
The equation of motion of a system is described by a Lagrange function
),,( tLL xx , leading to a set of differential equations of second order,
0
ii x
L
x
L
dt
d
, i = 1, 2, …, n
Consider the following transformation of coordinates,
),,( ),,( tt pxxx
(39)
where ),,(
tx
L
i
ip xx
H L
),,( ),,( i tt LH px xx-pxi
i
(40)
xi has to be expressed in terms of ),,( tpx
) , ( ),,(),,( , tpx tt Li pxpx xx-i
i
The Hamiltonian function
On one hand,
),,( tHH px
implies,
dH = dtt
Hdp
p
Hdx
x
Hi
ii
i
ii
(41)
On the other hand, the expression for ),,( ),,( i tLt pxi
iH xx-px given in (40)
implies,
dH = dtt
Lxd
x
Ldx
x
Ldpxxdp i
ii
i
iii
ii
i
ii
)(ix
L
dtd
ip
)( ipdtd
ip
Thus, the first and fourth sum-terms cancel out
dH = dtt
Ldxpdpx ii
ii
ii
(42)
From (41) and (42) one obtains,
),,( ),,( i tt LH px xx-pxi
i
) , ( ),,(),,( , tpx tt Li pxpx xx-i
i
Hamilton canonical equations (2n first-order equations) (43)
i
ip
Hx
i
ix
Hp
i = 1, 2, …, n and
(44) t
L
t
H
Notice if L does not depend on t explicitly, neither does H.
Recipe for solving problems in mechanics
i) Set up the Lagrangian ),,( tL xx .
ii) Obtain the canonical momenta ),,(
tx
L
i
ip xx
iii) Obtain ),,( ),,( i tLt pxi
iH xx-px in terms of x
and p
.
Example. Two dimensional motion of a particle in a central potential V( r
) =V(r).
L( ),,, rr )= T - V = 2
1m ( 222 rr ) - V(r).
Using the definition (39),
r
Lpr
= m r and
Lp = m 2r
Using the definition (40),
),,,( ),,,( r rrLrr pppprH -
ppr r
2
1m ( 222 rr ) V(r).
The expressions r
Lpr
and
Lp are inverted to
express r and in terms of ),,,( ppr
p
pp
p
mrm
2rr
2
1m ( 2
2
22r )()(mr
rm
pp )
),,,( pprrH = 2
22
r
2
2 mrm
pp V(r).
Using (30) one obtains the equation of motion for each of the 4 independent variables
pprr ,,, .
rp
Hr
=
m
pr
p
H
= 2
mr
p
r
Hpr
=
r
V
Hp r = 0
Properties of the Hamiltonian The significance of the Hamiltonian was already observed in Section 4.1C, in which,
for the case when 0
t
L the Hamiltonian constitutes a constant of motion. Now that
we have a more formal definition of the Hamiltonian, given in expression (40), we can re-state it the following manner:
t
H
dt
dH
(45)
Proof: dt
dH
t
Hp
p
Hx
x
Hi
ii
i
ii
- ip ix
The first and second sum-term cancel out, which proves the statement. That is,
(59)
a Hamiltonian that does not depend explicitly on time constitutes a constant of motion
H is a constant of motion if . (46)
L does not depend explicitly on t.
This follows from the two general results (44) t
L
t
H
and (45)
t
H
dt
dH
,
Sin 0
t
L implies 0
t
H, and, hence 0
dt
dH.
For a Lagrangian L=T-V where the potential V is independent
of x
and the kinetic energy T is homogeneous quadratic (47)
in ,x
then H is the total energy
2)( ii
i
xkT and L then has the form )( )( 2x
VxkLi
ii
One obtains, ix
L
= ii xk 2 and
i
ii
i i
i xkx
Lx 2)(2
= 2T
H = i
i
i x
Lx
L = 2T – L= 2T – ( T – V )= T + V .
H may be a constant of the motion but not the energy (if T is not homogeneous quadratic.) .
H may be the energy but not constant (if 0
t
L). .
H may be neither the energy nor a constant of motion.
In the Hamiltonian approach,
A mechanical system is completely specified at any time by giving all the
px
and coordinates.
That is, the state of a system is specified by a point ) ,( px
in phase space.
The task is to find how this point moves in time, i.e. the motion in phase space
The initial condition tell us where in phase space the system starts, but it is the Hamiltonian which tells us, though the canonical equations (43), how it proceeds from there.
( p
,x )
t=to
t
phase space
Alternatively, the Hamiltonian determines all the possible motions the system can perform in phase space, the initial conditions picking out the particular motion which is the solution to a particular problem.
4.2C Finding constant of motion before calculating the motion itself
a) Looking for functions whose Poisson bracket with the Hamiltonian vanishes It is possible to use the Hamiltonian to determine directly how a given dynamical
function varies along the solution-motion, even before calculating the motion itself. (For example, determining whether that quantity is a constant of motion or not.) To that effect, let’s consider a dynamical function F= F ) ,( t,px
and calculate its total
differential change with time,
dt
dF
t
Fp
p
Fx
x
Fi
ii
i
ii
ip
H
ix
H
-
t
F
x
H
p
F
p
H
x
F
iiiiii
dt
dF
t
F
x
H
p
F
p
H
x
F
i iiii
(48)
The first term on the right side is called the Poisson bracket between F and H (to be described in more detailed in the sections below).
i iiii x
H
p
F
p
H
x
FHF, (49)
In terms of the Poisson bracket, the time dependence of a physical quantity is expressed as,
dt
dF HF,
t
F
(48)
A quantity F that does not depend explicitly on time, will be a constant of motion if the Poisson bracket between F and the Hamiltonian H vanishes. Certainly, one would have to have a lot of intuition to figure out such a function F. We will see later, however, that there exist systematic methods to find just that. Here we just want to show that the possibility of finding constant of motion without solving the equations.
To see how this works, let’s take the familiar example of the two dimensional, symmetric simple harmonic oscillator. 2 For simplicity take k=1 and m=1. Then,
)()(),,,,(2
2
2
1
2
2
2
121212
1
2
1xxxxtxxxxL (49)
ii xtxxxxx
L
i
p
),,,,( 2121 for i= 1,2 (50)
),,( ),,( tL - xt i
i
i pH xxpx
),,,,( ),,,,( 212122112121 txxxxL -x xtppxx ppH
2211 pp x x )()(2
2
2
1
2
2
2
12
1
2
1xxxx
where we have to replace the ix in terms of the ip given in (50)
2211 pppp )()(2
2
2
1
2
2
2
12
1
2
1xxpp
),,,,( 2121 tppxxH )()(2
2
2
1
2
2
2
12
1
2
1xxpp (51)
Without finding the explicit solution, let’s show that expression (48) tell us that the angular momentum F,
12212121 ),,,,( pxpxtppqqF angular momentum (52)
Is a constant of the motion.
First, let’s calculate,
1212
1111
xxppx
H
p
F
p
H
x
F
2121
2222
xxppx
H
p
F
p
H
x
F
0
t
F
which gives
t
F
x
H
p
F
p
H
x
F
iiii
2
1i
=0
Accordingly, expression (48) gives,
0 dt
dF (53)
That is, without explicitly calculating the solution, we know that, for this problem, the angular momentum is a constant of motion.
b) Cyclic coordinates According to the definition given in Section 4.1C, a cyclic coordinate xi is one that does
not appear in the Lagrangian. The Lagrangian equation ii x
L
dt
d
x
L
then implies that
the generalized momentum pi =ix
L
is a constant of motion. But the Hamiltonian
equation i
ix
Hp
implies also that in this case 0
ix
H; that is,
a coordinate the is cyclic (i.e. absent in the Lagrangian) (54) will also be absent from the Hamiltonian
This conclusion can also be obtained from the definition of the Hamiltonian,
),,( ),,( i tt LH px xx-pxi
i
. Notice, H differ from – L by ipxi
i , which does
not involved the coordinates xi explicitly. Hamiltonian with full cyclic coordinates The solution of the Hamilton’s equation is trivial for the particular case in which the Hamiltonian does not depend explicitly on time (it is a constant of motion) and all coordinates xi are cyclic.
xi i=1, …, n does not appear in the Hamiltonian
Under those conditions all the conjugate momenta i
ix
Hp
will be constant.
iip
The Hamiltonian then may be written in the form
) , ... , ,( n21 HH
and the equation of motion for the coordinates x i will be,
) ( constantH
xi
i i
whose solutions are, ii txi
It is true that rarely occurs in practice that all the coordinates are cyclic (for, thus, taking advantage of the easy way to find the solutions.) However, a given problem can be described by different sets of coordinates. It becomes important then to find a systematic procedure for transforming from one set of variables to another set of variables where the solution is more conveniently tractable. That will be the subject of using canonical transformations, to be described in section 4.4. 4.2D The modified Hamilton’s principle: Derivation of the Hamilton’s equations from a
variational principle. Hamilton’s canonical equation can be obtained from a variational principle, similar
to the way the Lagrange equations were obtained in Section 4.1B above. However, the variations will be over paths in the ) ,( px phase-space, which has 2n dimensions, twice
the n dimensions of the x
configuration-space. Interestingly enough, the function inside the integral, upon which the variational principle will be applied, is again the Lagrangian L, but now considered as a function of ),, ,( t,ppxx .
) ( ),, ,( ,, tpxL Ht, i p-ppxx xi
i (55)
As a first step, let’s apply the variational principle to
),( pxS ) ≡
)(
)(
222
111
t,
t, ,
, , px
px
),, ,( t,L ppxx dt (56)
Inside the integral, we have just written ),, ,( t,ppxx for
simplicity, but ) , , , ( )()()()( t,tttt ppxx should be used
instead, respectively.
In applying the variational principle, we realized it is very similar to the case when we applied it in the configuration space. This time we just have more independent variables. The result is,
0x
L
x
L
dt
d
ii
and 0
p
L
p
L
dt
d
ii
, for i=1,2, …, n. (57)
Applying (57) to the Lagrangian given in (55), ) ( ),, ,( ,, tpxL Ht, i p-ppxx xi
i ,
ipix
L
,
ii x
H
x
L
-
0x
L
x
L
dt
d
ii
implies
ix
Hip
(58a)
On the other hand,
0
ip
L
,
i
i
i p
Hx
p
L
-
0p
L
p
L
dt
d
ii
implies
i
ip
Hx
(58b)
That is, we obtain in (58) the canonical Hamiltonian equations (43).
4.3 The Poisson bracket Expression (48) evaluates how a given dynamic quantity F= F ) ,( t,px
varies as a
function of time, while x
and p
evolves according to the Hamilton equations. The first
term of the right hand in (48) turns out to be an important expression in itself; it is
called the Poisson bracket of F and H.
In general, the Poisson bracket [S,R] of the dynamical variable S= S ) ,( px
with the
dynamical variable R= R ) ,( px
is defined as,
i iiii x
R
p
S
p
R
x
SRS, (55)
Poisson bracket of the dynamic quantities R and S
Note: Sometimes it will be convenient to express the Poisson bracket as
[S,R](x,p) (56)
to emphasize that the derivatives in the bracket are taken with respect to the variables ) ,( px
. For, it may happen that an (invertible) transformation of
coordinates
),,( tpxQQ
and ),,( tpxPP
may have taken place. Therefore the dynamical quantities will have a
dependence on Q
and P
, and the Poisson bracket of the S and R can be taken
with respect to the new variables,
[S,R](Q,P) (57)
In terms of the Poisson bracket, expression (48) can be expressed as,
dt
dF
t
FHF
],[ (58)
4.3A The Hamiltonian equations in terms of the Poisson brackets
In the particular case that F = x , expression (58) gives,
][][ H,xt
xH,x
dt
dxα
αα
α
But notice that ][ H,xα =
i ii
α
ii
α
x
H
p
x
p
H
x
x=
p
H
Therefore dt
dxα =p
H
= ][ H,xα
In the particular case that F=p, expression (58) gives,
],[],[
Hpt
pHp
dt
dp
But notice that ][ H,pα =
i ii
α
ii
α
x
H
p
p
p
H
x
p=
x
H
Therefore ][ H,pdt
dpα
α =x
H
Thus,
(59)
][ H,xdt
dxi
i
],[ Hpdt
dpi
i
Hamilton canonical equations (2n first-order equations)
i = 1, 2, …, n
constitutes an alternative way to express the Hamilton canonical equations.
4.3B Fundamental brackets
i iiii x
x
p
x
p
x
x
xxx
, = 0 for any 1, 2, ..., n
i iiii x
p
p
p
p
p
x
ppp
, = 0 for any 1, 2, ..., n
i iiii x
p
p
x
p
p
x
xpx
, = 0 for ≠
i iiii x
p
p
x
p
p
x
xpx
, = 1 for =
That is,
0, , )()( px,px, ][ ][ ppxx (60)
, )( px,][ px … ..
Remark. Notice, the result (60) appears to be trivial. It is. In fact, this is a property of the Poisson bracket itself, regardless of the existence of a Hamiltonian.
It turns out, however, that if the bracket of the variables px , were calculated with
respect to other arbitrary variables (Q,P), the value of )(, PQ,][ px would be, in general,
different than .
(61) For an arbitrary transformation (x,p) (Q,P)
, )( PQ,][ px (61)
But, for a particular type of transformation of coordinates, called canonical transformations (which are introduced in connection to Hamiltonian description of motion, to be described in detail in the following sections), the Poisson bracket has the
remarkable property of remaining constant, that is
, )( PQ,][ px , regardless of
the new canonical variables used to describe the motion. Hence the usefulness of the Poisson bracket; it helps to identify such important canonical transformations.
Here we derive the chain rule that relates the Poisson brackets evaluated with respect to different coordinates.
If S=S ) ,( PQ
and R=R ) ,( PQ
where ),,( tpxQQ
and ),,( tpxPP
i iiii
px,
x
R
p
S
p
R
x
SRS )(, =
i
)( )( i
β
βi
β
βα i
α
αi
α
α p
P
P
R
p
Q
Q
R
x
P
P
S
x
Q
Q
S
i
)( )( i
β
βi
β
βi
α
αi
α
α x
P
P
R
x
Q
Q
R
p
P
P
S
p
Q
Q
S
)
(
i
β
βi
α
αi
β
βi
α
α
i
β
βi
α
αi
β
βi
α
α
p
P
P
R
x
P
P
S
p
Q
Q
R
x
P
P
S
p
P
P
R
x
Q
Q
S
p
Q
Q
R
x
Q
Q
S
i
)
(
i
β
βi
α
αi
β
βi
α
α
i
β
βi
α
αi
β
βi
α
α
x
P
P
R
p
P
P
S
x
Q
Q
R
p
P
P
S
x
P
P
R
p
Q
Q
S
x
Q
Q
R
p
Q
Q
S
i
β
px,
βα
αβ
px,
βα
α P
RP,Q
Q
S
Q
RQ,Q
Q
S )()( ][][ (
+ ) ][][ )()(
β
px,
βα
αβ
px,
βα
α P
RP,P
P
S
Q
RQ,P
P
S
Thus,
For S=S ) ,( PQ
and R=R ) ,( PQ
where ),,( tpxQQ
and ),,( tpxPP
(62)
P
RP,P
P
S
Q
RQ,P
P
S
P
RP,Q
Q
S
Q
RQ,Q
Q
S
β
px,
βα
αβ
px,
βα
α
β
px,
βα
αβ
px,
βα
αβα
)()(
)()(
][][
][][
i i iii
px,
x
R
p
S
p
R
x
S
RS,)(
Notice, if a particular transformation of coordinates (x,p) (Q,P) satisfies,
)(],[ px,
αβ PQ = ,
)(],[ px,
αβ QQ = 0, and (63)
)(],[ px,
αβ PP = ,
then (62) gives )( px,
RS,
Q
R
P
S
P
R
Q
S
iiiii
. That is,
)( px,
RS, ) ( PQ,RS, Valid only for those particular trans- (64)
formation (x,p) (Q,P) that satisfy (63)
Thus, we have found that those transformation of coordinates (x,p) (Q,P) satisfying (63) are very special: the bracket of two arbitrary dynamical quantities R and S remain invariant, independent of whether we use (x,p) or (Q,P) to evaluate the bracket.
4.3C The Poisson bracket theorem: Preserving the description of the classical motion
in terms of a Hamiltonian a) Example of a motion for which there is not Hamiltonian to describe it
Given a Hamiltonian ),,( tH px
we say that the time evolution of the classical system
is generated by H according to the canonical equations (43) and (59). There are cases, however, in which no Hamiltonian generates a particular motion. Consider, for example, the following motion:
22 )( xpx (65)
22
2)(
)(21 xp 2x-
xp
xp
That there is no H whose associated canonical equation ends up in (60) can be seen by first requiring,
22 )( xp p
H x
22
2)(
)(21 xp 2x-
xp
x
x
H p
The first equation gives,
)(4)( 2222
xpxxp x
px
H
and the second equation gives,
])()(2
1[ 22
2
2
xp 2x- xp
x
p
xp
H
xp 4x- xp
x - )(
)(21 2
22
The fact that
xp
H
px
H
22
(66)
implies that such a function H does not exist.
b) Change of coordinates to attain a Hamiltonian description
It is interesting to note in the example above that under a proper change of coordinates (x,p) (Q,P), a Hamiltonian function K can be found such that
P
K Q
and
Q
K- P
. The following transformation will do the trick.
xQ (67)
22 )( xp P
The equations of motion for Q and P are,
xQ according to (65)
22 )( xp
= P
)2()( 2 xxpxp2 P
= 2P1/2
( p +2QQ ) according to (65)
= 2P1/2
( 22
2)(
)(21 xp 2x-
xp
x
+ 2QQ )
= 2P1/2
( PQ 2 - P
Q
1/221 2QQ )
But PQ
= 2P1/2
( P
Q
1/221 )
Q P
In short, the motion described, in the x,p coordinates, as 22 )( xpx
22
2)(
)(21 xp 2x-
xp
xp
for which there is not a Hamiltonian,
is alternatively described in the new coordinates Q , P by the following equation of motion,
PQ (68)
Q P
These equations do have a Hamiltonian. Indeed,
setting P P
K Q
, gives K(Q,P) = (1/2)P2 + f (Q)
setting Q Q
K P
, gives K(Q,P) = (1/2)Q2 + g (P)
Therefore,
K(Q,P) = (1/2) Q2 + (1/2) P2 (69)
c) The Poisson bracket theorem The above remark indicates that, when a Hamiltonian cannot be found, it may be
that we are using the “wrong” coordinates. “Wrong” in the sense that it does not exist a Hamiltonian K to describe the motion in terms of the canonical equations (43) and (59). The following Poisson bracket theorem comes handy, then, as a guidance to recognize situations in which the motion, in the coordinates being used to describe it, can be generated by a Hamiltonian.
(70)
Let )(,)( tt px
be the time development of a system on phase space.
This development is generated by a Hamiltonian ),,( tH px
if and only if every pair of dynamic variables ),,( tR px
, ),,( tS px
satisfies the
relation
], [ ] [][dt
dSR S,
dt
dRSR,
dt
d
An outline of the proof is given in the Appendix-1, at the end of this chapter. This theorem is used to verify whether or not a given motion is a Hamiltonian motion. If we knew x=x(t) and p = p(t), we would apply (70) with the choice of R=q and S=p. If (70) were not fulfilled, then there will be no Hamiltonian to describe such a motion.
4.4 Canonical Transformations In many occasions, it may be convenient to make a transformation of coordinates
(x,p) (Q,P). The motivation may not be necessarily to simplify the mathematical burden in solving the Hamiltonian equations in the new coordinates; instead, more often it is to identify more clearly those physical quantities that remain constants
throughout the motion (even without solving the equations of motion.) In other occasions, the transformation of coordinates allows a better interpretation of classical mechanics formalism as a point of departure for elaborating a quantum theory. The latter is more pertinent to this course.
The example given in the previous section illustrate, however, that, if we want to keep a Hamiltonian formalism to describe the motion, we have to be careful when making a proper change of coordinates. Otherwise we may end up with no Hamiltonian associated to that motion when described in those new coordinates. In this section we study those types of transformations of coordinates that allow keeping the description of the classical motion within a Hamiltonian formalism, i. e. its description according to the canonical equations (43) and (59). They are called canonical transformations. We will find that the systematic procedure for generating such transformations, involves the participation of four types of so called generating functions. Each generating function produces a corresponding type of canonical transformation. In the course of constructing a consistent formalism for obtaining canonical transformations, a fortunate turn of events occurs. It turns out that, finding the proper generating function (to obtain a canonical transformation of specific characteristics) is equivalent to finding the solution to the canonical Hamiltonian equations! (This will be shown in Section 4.5C below.) Further, a particular canonical transformation will lead to a new Hamiltonian that is identically equal to zero; the equation that such a generating function must satisfy is the celebrated Hamilton Jacobi equation (this will be addressed in Section 4.7). In the subsequent chapters, when elaborating a quantum mechanics formalism, we will require that the quantum mechanics equation should have the Hamilton Jacobi equation as a limit. The latter justify our current effort for attaining a god understanding of the Hamilton Jacoby equation. We start by addressing the concept of canonical transformations first. Transformation of coordinates Consider that a classical motion is described by
)( ,)( tt px (71)
for which a Hamiltonian exists; i.e. the changes of )(tx
and )(tp
are governed by a
Hamiltonian H:
i
ip
Hx
and
i
ix
Hp
(72)
Consider a transformation of coordinates,
),,( ),,( tt P
Qpx (73)
),,( ; ),,( tpxtpxQQ
PP
As )(tx
and )(tp
change, the variables )(tQ
and )(tP
also change accordingly.
However, nothing ensures that the time evolution of the latter variables preserves the
Hamiltonian formalism given in expression (43). That is, it doesn’t always exists a
function K such that i
iP
KQ
and
i
iQ
KP
.
Those particular transformations that allow preserving the Hamiltonian form of the equation of motion are called canonical transformation. In this section we address,
a method to generate such transformations, and
inquire how to select a canonical transformation that renders a Hamiltonian K that is a constant function, for, in such a case, the equation of motion becomes obviously very simple.
( p
,x )
t=to
t
phase space ( P
,Q )
t=to
t
phase space
H
Transformation
Figure 3. For
),,( ; ),,( tpxtpxQQ
PP
4.4A Canonoid transformations (i.e. not quite canonical)
Preservation of the canonical equations with respect to a particular Hamiltonian)
Consider a Hamiltonian H = H ).,,( tpx
An invertible transformation of variables ),,( ),,( tt P
Qpx such
that the time evolution of the new variables )(tiQ
and )(tP
(74)
preserve the form of the Hamiltonian equations,
i.e. there exists a function K such that i
iP
KQ
and
i
iQ
KP
,
is called canonoid (i.e. not quite canonical) with respect to H.
. Note: It turns out, a transformation that is canonoid with respect to a given Hamiltonian need not be so with respect to another.
4.4B Canonical transformations
Definition A transformation that is canonoid with respect to all Hamiltonians (75) is called canonical.
Canonical transformation theorem3 Let ),( px
be a set of general coordinates on phase space. The Poisson brackets of any
two dynamic variables R and S will be specified as,
i iiii
px
x
R
p
S
p
R
x
SS
),(R,
Consider an invertible transformation
),,( ; ),,( tpxtpxQQ PP
The following three statements are equivalent:
a) The transformation ),,( ; ),,( tt pxpxQQ PP is canonical. (76)
b) There exists a nonzero constant z such that any dynamical variables R and S satisfy,
),(),(R, R,
pxQSzS
P (77)
(That is, the Poisson bracket is practically independent of the coordinates used to calculate it.)
The transformation ),,( ; ),,( tt pxpxQQ PP is canonoid with
respect to all quadratic Hamiltonians of the form,
H= C +
n
pcxc1
)'(
+
+
n n
pppxxx1 1
)"'(2
1
2
1
(78)
where = , ' = ', ” = ”
An outline of the proof is given in the Appendix-2.
Canonical transformation and the invariance of the Poisson bracket In short, the theorem above establishes that,
A transformation is canonical if and only if (79)
It preserves the Poisson bracket (to within a constant factor z.)
4.4C Restricted canonical transformations A canonical transformation ),,( ; ),,( tt pxpxQQ PP is called (80)
restricted-canonical transformation if in expression (78) z=1.
That is, ),(),(R, R,
pxQSS
P.
Note: In the literature the restricted canonical transformations are often simply called canonical transformations.
4.5 How to generate (restricted) canonical transformations
4.5A Generating functions of transformations In the remaining of the notes, whenever referring to canonical transformations, we
will assume that they are restricted (z=1); that is ),(),(R, R,
pxQSS
P. A way to
generate restricted canonical transformations will result along the way of attempting to classify them. Toward this end, let’s analyze first the following expression
ii P i
i
i
i Qpx - (81)
where the variables correspond to the transformation (73),
),,( ; ),,( tpxtpxQQ PP (82)
It turns out, when (81) is written as a function of ),,( tpx
it reveals much about the
transformation. It is found that when the transformation (82) is (restricted) canonical,
expression (81) becomes an exact total derivative of a scalar function F. The latter is then used to generate the transformation
The strategy is to show that there exists a function F that allows express (81) in the following alternative form,
ii P i
i
i
i Qpx - = else ...t
Fp
p
Fx
x
Fi
i
i
i
ii
(83)
To this end, let’s expand the left side in terms of the old coordinates,
ii P i
i
i
i Qpx - =
= ik P )(t
Q
p
Q
x
Q i
k
i
k
ii
i
i pxpx
ki k
k-
iQ
)PPP( imimi
i
i
i i
m
mm i
i
i
m
i
it
Q
p
Q
x
Qpxpx -
i
i
m i
m
ii m
i
i
mi
t
Q
p
Q
x
Qpxp iimm P)P()P( (84)
i i
The next step is then to identify, term by term, expressions (83) and (84).
Appendix-3 outlines the demonstration that there exists such a scalar function F such that,
im
m
i
mi
x
F P
x
Q p
im
m
i
m
p
F P
p
Q
and
KHt
FP
t
Q
i
ii
(85)
Further, replacing (85) in (84),
ii P i
i
i
i Qpx - = t
Fp
p
Fx
x
Fi
i
i
i
(
ii
+ H - K)
Or
H - i
i
i px -
KPQ i
i
i =
t
Fp
p
Fx
x
Fi
i
i
i
ii
H - i
i
i px -
KPQ i
i
i =
dt
dF (86)
What is remarkable is that the left side is an exact differential. The function F is called the generating function of the transformation for, as we will see, once F is given the transformation equations (82).
4.5B Classification of (restricted) canonical transformations Expression (86) suggests that, in order to effect the transformation between the two
sets of canonical variables, F must be a function of both the old and the new variables. F is a function of 4n variables plus the time. But only 2n of these are independent, because the two sets ) , P(Q and ),( px are connected by the 2n transformation
equations (82). Accordingly, there are four potential forms to express F, which will depend on the circumstances dictated by the specific problem:
),(1 t ,F Qx = ) ,( ) ,( t ,F Qxpx
),(2 t ,F Px = ), ,( ) ,( t F Pxpx ,
),(3 t ,F Qp = ) ,( ) ,( t ,F px Qp
),(4 t ,F Pp = ) ,( ) ,( t ,F px p P
Case 1: Assume the function ),(1 t ,F Qq is given.
In this case, (86) states,
H - i
i
i px -
KQ i
i
iP = ),(1 t ,dt
dFQx
=t
FQ
Q
Fx
x
Fi
i
i
i
111
ii
(87)
Since the xi and the Qi are independent, then the coefficients of ix and iQ should be
equal,
),(1 t ,x
F
i
ip Qx
i= 1, 2, … , n (88a)
These are used to solve for the n variables Qi as a function of ),( px
),(1P t ,Q
F
i
i Qx
i= 1, 2, … , n (88b)
Since the Qi have been determined in (88a), this expression gives the n variables Pi as a function of ),( px
which leaves (87) with,
t
FHK
1 (88c)
Case 2: Assume the function ),(2 t ,F Px is given.
In this case, (86) states,
H - i
i
i px -
KQ i
i
iP = ),(2 t ,dt
dFPx
=t
FFx
x
Fi
i
i
i
222 P
P
ii
(89)
On the left side, we expand the term i
i
iQ P . For convenience we write it as m
m
mQ P ,
H - i
i
i px -
KQ m
m
mP = ),(2 t ,dt
dFPx
=t
FFx
x
Fi
i
i
i
222 P
P
ii
(89)’
where we will replace PP
mmi
i
i
i i
m
Qx
x
i
m
m
mQ P = m
m
i
i
i
i i
Qx
x
QP ) P
P ( mm
i
Inverting the order of the summation
m
m
mQ P = P P
PP mm
i
im
ii
im
m i
Qx
x
Q m
(90)
Replacing (90) in (89)’
H - i
i
i px -
i
im
ii
im
m i
Qx
x
QP )
P ( ) ( PP mm
m
+ K = ),(2 t ,dt
dFPx
= t
FFx
x
Fi
i
i
i
222 P
P
ii
(91)
Since the xi and the Pi are independent in (91), then the coefficients of ix and iP
should be equal,
),( P 2m t ,x
F
x
Q
i
m
m i
ip Px
PP
mm
i
Q
m
= ),(P
2 t ,F
i
Px
(92)
K - H = t
F
2
It is not straightforward to visualize that from the first equation the Pi can be solved as a function of ),( px , since the Qm are involved there. Hence, we perform an extra step.
First, notice that the first equation in (92) can be written as,
),( P 2m t ,
x
FQ
x i
m
mi
ip Px
)P( m2 m
mi
i QFx
p
(93a)
Also notice in the second equation of (92),
m
i
m
ii
m QQQ
mmm P
PP
P
P
)P( mmm
mm
i
m
im
m
PP
m
im
i
P
P m
m
Hence, the second equation in (92) can be expressed as,
im
P
)P(Q
Q
i
m
m
= ),(P
2 t ,F
i
Px
iQ =
m
)P(P
m2 m
i
QF (93b)
In term of the function F2’,
m
)P( ' m22 mQFF ,
expressions (92a) and (92b) give,
i
ix
Fp
' 2
iQ = i
F
P
'2
(94)
K - H = t
F
' 2
At the end, there will be no interest in the function F2. We already found a F2’ that, if expressed in terms of (x,P) as independent variables, it will generate a canonical transformation. Hence, let’s just rename the F2’ as F2. In summary
),( 2 t ,x
F
i
ip Px
, iQ = ),(
P
2 t ,F
i
Px
, K - H =
t
F
2
solve for Once the Pi are known,
this gives
),,(PP t ii px ),,( t QQ ii px
Invert them to obtain
),, t P x(Qx ),, t K P (Q = ),,( t H px +t
F
' 2 ),( t ,Px
),, t P p(Qp
),, t K P (Q = ), ,( ),,),, t H t t P (QP (Q px +
+t
F
' 2 ),( ),, t ,t PP (Qx
4.5C Evolution of the mechanical state viewed as series of canonical transformations i) The generator of the identity transformation Consider the generating function
i
x i2 P),( ixt ,F P (95)
Let’s find out the canonical transformation it generates. Using (94),
i
i
ix
Fp P 2
(96)
iQ = i
i
xF
P
2
We find the new coordinates ) ,( PQ are the same old coordinates ),( px . That is, the
function in i
x i2 ),( Pxt ,F iP generates the identity transformation.
),( px I
) ,( PQ
ii) Infinitesimal transformations Consider the generating function
i
x i2 ),( Pxt ,F iP + ),( t ,G px (97)
where is an infinitesimal number, and ),( t ,G px an arbitrary function (to be specified
later.)
Due to the small value of , and since i
iPxi generates the identity transformation, we
expect the new coordinates ) ,( PQ will differ from the old ones ),( px also by
infinitesimal values; that is,
iQ = iix (98)
iP = iip
Let’s figure out the values of i and i .
Applying (94) to (97) gives,
ii
ix
G
x
Fp
i
2 P (99a)
iQ = ),(PP
)(2 t ,Gx
F
i
i
i
Ppx
based on (99a) one obtains
),( )( t ,Gp
xi
i Ppx
ip
iP
),( )( t ,Gp
xi
i Ppx
(99b)
In short, we have obtained:
The function i
x i2 ),( Pxt ,F iP + ),( t ,G px generates
the transformation
i
ix
Gp
Pi (100)
iQ i
ip
Gx
iii) The Hamiltonian as a generating function of canonical transformations
Notice in (100), if G is the Hamiltonian ),( t ,H px , and is chosen to be the incremental
time differential dt , one obtains,
i
ix
Gp
Pi =
i
ix
Hdtp
)(
Using the Hamiltonian equations (43)
= )( )( ii pdtp
But )( dtpi is the increment of p due to the motion
= ii dpp
iQ i
ip
Gx
=
i
ip
Hdtx
)(
Using the Hamiltonian equations (43)
= )( )( ii xdtx
= ii dxx
That is, we have obtained:
(70)
The function i
x i2 ),( Pxt ,F iP + (dt) ),( t ,H px generates the
transformation ),( px ) ,( PQ
ii dxx Q i (101)
ii dpp Pi
where dxi and dpi are, respectively, the changes in xi and pi due to the motion governed by the Hamiltonia H.
Notice in (101) that, basically, the Hamiltonian can be viewed as the generator of the transformation. It transforms the value of the coordinates
),( px at the time t, to the value of those coordinates at the time t +dt.
This result is very interesting. It tells us that the evolution of the state of
motion ) ,( )()( tt ii px can be viewed as a series of canonical transformations
generated by the function H, (as the result applying (101) successively one after another differential time dt.)
iv) Time evolution of a mechanical state viewed as a canonical transformation Symbolically, the result in (101) can be expressed this way:
Defining )( t't,F i
iPxi + (t’-t) ),( t ,H px (102)
x (t), p(t) )( t't,F
x(t’), p(t’) )( t"t,F
x(t”), p((t”) , …. , (103)
Evolution of the classical state x (t), p(t) over time. The Hamiltonian in (102) constitutes the generator of the state’s evolution with time.
Expression (103) also hints an approach, at least conceptually, of how to attain a solution of the equation of motion. In effect, since the successive applications of
canonical transformations is a canonical transformation, finding the solution x (t), p(t)
of the Hamilton equations can be view as the task of finding a canonical transformation
that takes the initial conditions x(t0), p(t0) (old coordinates) to the values of the state at
the time t, x (t), p(t) (the new coordinates).
Canonical transformation
x(t0), p(t0) )( 0tt,F
x(t), p(t) (104) old coordinates new coordinates
4.6 Universality of the Lagrangian 4.6A Invariant of the Lagrangian equation with respect to the coordinates used in the
configuration space It was stated in the sections 4.1B above that the Lagrangian equations
0x
L
x
L
dt
d
ii
(105)
have the particular interesting property of being independent of the particular coordinates used in the configuration space. Regardless of the coordinates being used, the Lagrangian equations look the same. From each new coordinates we would be able to obtain a corresponding Hamiltonian, by following the procedure of Section 4.2B. Such a statement may appear to be at odds with the concepts outlined in Section 4.4, where we had to be careful in not making the wrong transformation of coordinates, otherwise we would end up with a no Hamiltonian describing the system. The transformation had to be canonical. The following example helps clarify the issue. Consider the generating function,
m
mm t ft ,F P)(),(2 xx P (106)
where the fi are arbitrary functions.
Using (94),
iQ = )(P
2 t fF
i
i
x
(107)
ii
ix
Fp
2 =
m
m
i
m
x
t fP
)(x
That is, the new coordinates Qi depend only on the old coordinates and time, but do not involve the old momenta. Such a transformation is then an example of the class of point transformation, done in the configuration space, which led to the consideration of the invariant of the Lagrangian equation with respect to a transformation of coordinates. In this context, such a point transformation (107) are canonical, and therefore preserve the Hamiltonian description of the motion. We have therefore a view of the invariance of the Lagrangian from a canonical transformation of coordinates perspective.
4.6B The Lagrangian equation as an invariant operator
In Section 4.2D, the application of the modified Hamilton’s variational principle to the generalized Lagrangian ),, ,( t,L ppxx led to
0x
L
x
L
dt
d
ii
and 0
p
L
p
L
dt
d
ii
, for i=1,2, …, n (106)
For simplification purposes in this section let’s use, ζpx ) ,(
So we express the Lagrangian as,
),( t,L ζζ (107)
And the equation (106) take the compact form,
0 ),(
t,
dt
dL
ii
ζζ
, for i=1,2, …, 2n (108)
For new coordinates obtained through a canonical transformation
),,( ),,( tt P
Qpx
which in our new notation will be expressed as, ),( ),( tηtζ (109)
The application of the variational principle to the Lagrangian
),( t,L' ηη (110)
will lead to
0 ),(
t,
dt
dL'
ii
ηη
, for i=1,2, …, 2n (111)
On the other hand, expression (86) tell us that
)( t,,H - i
i
i px px -
)( t,,KPQ i
i
i PQ = dt
dF
or, in the new notation,
),( ),( dt
dFt,t, 'LL ηη-ζζ (112)
Now come the time to highlight further the importance of expression (112). It turns out that replacing (112) in (108), one obtains,
0 ),(
t,
dt
dL'
ii
ηη
(113)
That 0 ),(
t,
dt
dL
ii
ζζ
and 0 ),(
t,
dt
dL'
ii
ηη
are the same is
remarkable.
It was obtained not obtained because 0 ),(
t,
dt
dL
ii
ζζ
and
0 ),(
t,
dt
dL'
ii
ηη
are equations for the same set of motion in the
phase-space.
It was obtained only because L and L’ differ by a function dt
dF
It suggest that
iidt
d
is a kind of operator independent of the particular
coordinates being used. For, if the Lagrangian is expressed as ),( t,L ζζ or
),( t,L' ηη , the application of the operator
iidt
d
renders the
description of the same set of motion in the phase-space.
4.7 The Hamilton Jacobi equation
The Hamilton principal function We have been envisioning ways to obtain simple ways to solve the Hamiltonian
equations. In one case, we alluded to the convenience of finding a systematic way to transform from one set of coordinates to another set of coordinates in such a way that the new Hamiltonian ends up having all the coordinates xi 1=1, 2,…,n being cyclic. Another approach was outlined in Section 4.5D, with a canonical transformation relating the old coordinates (taken as the state variables at a given time t) to the new coordinates (taken as the state variables at a time t’.)
We will follow this latter approach for being more convenient, since it applies even when the Hamiltonian depends on time (i.e. it is not a constant of motion.) The outlook will have a slight variation. We will be looking for a canonical transformation F such that
it transform the coordinates x(t), p(t) to a new set of coordinates x(t0), p(t0) (the latter are constant values of the initial conditions). In our custom terminology,
( x(t), p(t) ) )( 0tt,F
( Q , P ) (114)
( x(t0) , p(t0) )
( x0 , p0 )
New coordinates constant in time
H(x , p ) )( 0tt,F
K( Q, P)
One can automatically ensure that the new variables (Q , P) are constant in time by requiring that the new Hamiltonian K( Q, P) be identically zero!, for then the equations of motion are,
0P
i
KQi
(115)
0Q
P
i
Ki
K is related to the old Hamiltonian H and the generating function by,
t
FHK
which will be zero if,
0 ) , ,(
t
FtH px (116)
It is convenient to choose ),( Px as the independent coordinates; that is,
),(2 t ,F Px = ), ,( ) ,( t F Pxpx (117)
which gives the
i
ix
Fp
2 ),( t ,Px (118)
iQ = i
F
P
2
),( t ,Px
Equation (116) takes the form,
0 , , .... , , 2
n
2
1
2 )(
t
F t
x
F
x
FH x
Given H, this constitutes an equation for F2. It is customary to denote the solution of this equation by S, which is called the Hamilton’s principal function.
0 , , .... , , )(n1
t
S t
x
S
x
SH x (119)
Hamilton-Jacobi Equation ),( t ,SS Px with P = constant
Hamilton’s principal function
When solving (119), the n constants of integration can be taken as the Pi s.
Expression i
ix
Sp
),( t ,Px evaluated at t0 gives a relationship between the
constant value of P and the initial conditions x0 , p0.
Then using expression (118), iQ =i
F
P
2
),( t ,Px , one has a relationship to find the
Qis. In particular the Qis can be obtained by evaluating this expression at t0.
Inverting iQ =i
F
P
2
),( t ,Px , one obtains,
) ,( t ,PQxx (120)
which gives the the coordinates as a function of time and the initial conditions. In short, when solving the Hamilton-Jacobi equation, we are at the same time obtaining a solution of the mechanical problem
Further physical significance of the Hamilton principal function Let’s evaluate the total derivative of ),( t ,SS Px .
dt
dS
t
Sx
x
Si
i
i
because the Pis are constant .
Using expression (118), i
ix
Sp
),( t ,Px , gives
dt
dS
t
Sxp ii
i
Using the Hamilton Jacobi equation 0 , , )(
t
S tH px , o
dt
dS Hxp ii
i
The term on the right iis the Lagrangian )( , , tL px ; thus,
dt
dS L (121)
L is then an indefinite time integral of the Lagrangian,
dt' t't't'LS , , )( )()( pxt
(122)
Appendix-1 Poisson bracket theorem
Let (t)(t) ) , ( px
be the time development of a system on phase space. This
development is generated by some Hamiltonian ),,( tH px
if and only if
every pair of dynamic variables ),,( tR px
, ),,( tS px
satisfies the relation
],[],[],[ SRSRSRdt
d (1)
Outline of the proof:
H implying (1) is straightforward. Hint: The existence of a Hamiltonian implies that any variable F changes
according to dt
dF
t
FHF
],[ ). Use the last expression with F= [R,S].
That (1) implies the existence of a Hamiltonian governing the variations of (t)x
and (t)p
is more elaborated.
The variable (t)x , for example, will have the form,
(t)x = (t)(t) ) , ( px
= 1, 2, …, n
Similarly,
(t)pβ = (t)(t) ) , ( px
= 1, 2, …, n
(Below we demonstrate that (1) implies
x
=
p
)( )
We want to prove and can be obtained from a function H such that
pH/ and xH/ .
We need to demonstrate the existence of H.
Note: It may be sufficient to demonstrate that
x
=
p
)( (2)
For it will ensure not contradicting the fact that, if H existed, it would
have to satisfy xp
H
px
H
22
Since, regardless of the existence of a Hamiltonian or not,
, ][ px , we
start with the following result: 0],[ pxdt
d. Expression (1) implies
],[],[],[0 pxpxpxdt
d
Using x = and p = one obtains,
],[],[0 xp
i iiii x
p
pp
p
x
+
i iiii xp
x
px
x =0
x
+
p
=0
or
x
=
p
)(
In other words, (2) does not contradict the quest for finding a function H
that satisfies pH/ and xH/ .
________________________________________________________________________ Appendix-2 Canonical transformation theorem
Let ),( px
be a set of general coordinates on phase space. The Poisson brackets
of any two dynamic variables R and S will be specified as,
i iiii x
R
p
S
p
R
x
SS
),(R,
px
Consider an invertible transformation
),,( ; ),,( tt pxpxQQ
PP
The following three statements are equivalent:
a) The transformation ),,( ; ),,( tt pxpxQQ
PP is canonical.
b) There exists a nonzero constant z such that any dynamical variables R and S satisfy,
),(),(R, R,
pxQ
SzS P
(1)
(That is, the Poisson bracket is practically independent of the coordinates used to calculate it.)
c) The transformation ),,( ; ),,( tt pxpxQQ
PP is canonoid with
respect to all quadratic Hamiltonians of the form,
H= C +
n
pcxc1
)'(
+
n n
pppxxx1 1
)"'(2
1
2
1
(2)
where = , ' = ', ” = ”
Proof:
That b) implies a) is a bit straightforward. All it needs to be done is to show
),(),(),( ],[],[],[ PPP
QQQ SRSRSRdt
d . For, according to the Poisson bracket
theorem (shown above), the latter implies that the evolution of )(tQ
and )(tP
is generated by a Hamiltonian.
From b:) ),(),( R, ],[pxQ
Szdt
dSR
dt
dP
),(R,
px
Sdt
dz
Assuming that the evolution of ),( px
is governed by
a Hamiltonian, Poisson bracket theorem implies,
} ],[],[ {R, ),(),(),( pxpxpx
SRSRzSdt
dz
Using b) again,
],[],[ ),(),( PP
QQ SRSR
That c) implies b) is more elaborated.
Exploiting the fact that ),,( ; ),,( tt pxpxQQ
PP is canonoid with respect to
any quadratic Hamiltonian, we will figure out first the value of
),(],[ P
Q
xx = ? ,),(],[ P
Q
px =?, and ),(],[ P
Q
pp =?,
which then be used in the result for ),(],[ P
QSR obtained in (36).
Lets’ start considering the case that ),,( ; ),,( tt pxpxQQ
PP
is canonoid with respect to H=C (constant).
),,( tH px
constant implies,
0
ii
ip
const
p
Hx and 0
ii
ix
const
x
Hp (3)
Since ),,(),,( ; tt pxpxQQ
PP is canonoid with respect to the
Hamiltonian ),,( tH px
=const, it means (according to the Poisson bracket
theorem) that there exist a Hamiltonian (K= ),( PQK ) according to which a
quantities like ),(],[ P
Q
xx and ),(],[ P
Q
px would evolve in time as,
),(),(),( ],[],[],[ PPP
QQQ
xxxxxxdt
d
),(),(),( ],[],[],[ PPP
QQQ
pxpxpxdt
d
),(),(),( ],[],[],[ PPP
QQQ
ppppppdt
d
But according to (40) all the 0ix and 0ip , which gives
0],[ ),( P
Q
xxdt
d;
0],[ ),( P
Q
pxdt
d; and (4)
0],[ ),( P
Q
ppdt
d
Let’s put ),(],[ P
Q
xx , ),(],[ P
Q
px , and ),(],[ P
Q
pp as a
function of ),,( tpx
),(],[ P
Q
xx ≡ f ),,( tpx
(5)
),(],[ P
Q
px ≡ g ),,( tpx
),(],[ P
Q
pp ≡ s ),,( tpx
For a Hamiltonian H ),,( tpx
=const,
0],[ ),( px
Hf , 0],[ ),( px
Hg , 0],[ ),( px
Hs ; and
the time variation of f, g, and s, are given by,
ft
Hffdt
d
),(],[ px
= ft
gt
Hggdt
d
),(],[ px
= gt
st
Hssdt
d
),(],[ px
= gt
Since, according to (4), 0 fdt
d, 0 g
dt
d, and 0 s
dt
d, one
obtains,
0],[ ),(
P
Q
xxt
ft
; .
0],[ ),(
P
Q
pxt
gt
; and .. (6)
0],[ ),(
P
Q
ppt
st
…
So, neither ),(],[ P
Q
xx , ),(],[ P
Q
px
nor ),(],[ P
Q
pp depend explicitly on the
time variable. . Note: Notice that, although we have used the Hamiltonian H=constant to
obtain the result (6), neither of the values ),(],[ P
Q
xx , ),(],[ P
Q
px , or
),(],[ P
Q
pp depend on the particular Hamiltonian used. They depend
only on the transformation coordinates.
Consider now the Hamiltonian H=
n
pcxc1
)'(
This implies
i
i
i cp
Hx '
and i
i
i cx
Hp
(7)
Since ),,(),,( ; tt pxpxQQ
PP is canonoid with respect to this
Hamiltonian H, it means (according to the Poisson bracket theorem) that
there exist a Hamiltonian (K’= ),(' PQK ) according to which a quantities like
),(],[ P
Q
xx , ),(],[ P
Q
px and ),(],[ P
Q
pp would evolve in time as,
),(),(),( ],[],[],[ PPP
QQQ
xxxxxxdt
d ,
),(),(),( ],[],[],[ PPP
QQQ
pxpxpxdt
d , and
),(),(),( ],[],[],[ PPP
QQQ
ppppppdt
d
Using (7),
0],'[],[ ),(),( PP
xcxx ;
0],[],[ ),(),( PP
cxpx ; and
0],[],[ ),(),( PP
cppp
Thus,
0],[ ),( fdt
dxx
dt
d P
Q
0],[ ),( gdt
dpx
dt
d P
Q
0],[ ),( sdt
dpp
dt
d P
Q
Hence, in terms of the Hamiltonian H=H ),,( tpx
ft
Hffdt
d
),(],[0 px
gt
Hggdt
d
),(],[ 0 px
st
Hssdt
d
),(],[ 0 px
Using (6),
0],[ ),( px
Hf
0],[1
)'(
n
ipcxc iiiif
011
],['],[
n
i
ii
n
i
ii pfcxfc
011
'
n
k ki
n
k kk x
fc
p
fc
This should be true for all arbitrary values of ck and c’k, which implies,
0
kx
f and 0
kx
f for k = 1, 2, …. , n
Similarly, (8)
0
kx
g and 0
kx
g for k = 1, 2, …. , n.
From (6) and (8), the quantities ,
),(],[ P
Q
xx , ),(],[ P
Q
px and ),(],[ P
Q
pp (9)
do not depend on the ),,( tpx
variables. .
That is, they are constant values. .
Now we need to prove that all those constant values are of the same magnitude │z│. To that effect consider the Hamiltonian
H =
n n
pppxxx1 1
)"'(2
1
2
1
)""'(2
1
2
1
ppxp
Hx iii
i
i
)"'(
px ii (10)
)'(
pxx
Hp ii
i
i
)'((
px ii (10)’
Since ),,(),,( ; tt pxpxQQ
PP is canonoid with respect to this
Hamiltonian H, it means
),(),(),( ],[],[],[ PPP
QQQ
xxxxxxdt
d =0
),(),(),( ],[],[],[ PPP
QQQ
pxpxpxdt
d =0
),(),(),( ],[],[],[ PPP
QQQ
ppppppdt
d =0
(All these quantities are equal to zero, according to (9).)
Using (10), ),(),( ],[],[0 PP
xxxx implies,
),(
),(
)]"'(,[
]),"'([0
P
P
Q
Q
kkkk
k
kkkk
k
pxx
xpx
),(),(
),(),(
)],["],['(
],["],['(
PP
PP
kkkk
k
kkkk
k
pxxx
xpxx
0 = ('F) + (F') - (G”) + (G”) (11)
’=0 implies (G”) = (G”) ”=I implies G is symmetric
”=0 implies (’F) = (F’)’=I implies F in symmetric
Similarly, using (10) ),(),( ],[],[0 PP
pxpx implies,
),(
),(
)]'((,[
]),"'([0
P
P
Q
Q
kkkk
k
kkkk
k
pxx
ppx
),(),(
),(),(
)],['],[
],["]['(0
PP
PP
kkkk
k
kkkk
k
pxxx
pppx
0 = (’G) + (”S)- (F)- (G’) (12)
’=”=0 implies F = 0
=’=0 implies ”S=0; ”=I implies S=0
=”=0 implies ’G =G’ ; ’=I implies G=G
Using (10), ),(),( ],[],[0 PP
pppp implies
),(
),(
)]'((,[]),'(([0 PP
kkkk
k
kkkk
k
pxpppx
),(
),(
),(),(
)],['],[
],['],[0
PP
PP
kkkk
k
kkkk
k
ppxp
pppx
0 = (G) + (’S)- (G) + (S’) (13)
’=0 implies (G) = (G); =I implies G is symmetric
=0 implies ('S) = -(S’); '=I implies S= -S =0
From (11), (12) and (13)
F=0 or f ≡ ),(],[ P
Q
xx = 0
S=0 or s ≡ ),(],[ P
Q
pp = 0 (14)
G symmetric g≡ ),(],[ P
Q
px = g
We can further determine the g values for ≠.
For example:
Since the matrix is arbitrary, let’s consider one in which only 13=31 are the only elements ≠0.
......00
...
...000
...
...........
...
...
...
.........0
...00
...000
...00
133112311131
331332133113
333231
232221
131211
31
13
ggg
ggg
ggg
ggg
ggg
G
Expression (13) requires (G) = (G)which implies
g11= g33 and g32= g3n =0 for n>3
We arrive to the conclusion then that
g = const (15)
The constant value depends on the particular transformation used, and it is referred to as z in the text of the theorem.
Thus,
F=0 or f ≡ ),(],[ P
Q
xx = 0
S=0 or s ≡ ),(],[ P
Q
pp = 0 (16)
G symmetric g≡ ),(],[ P
Q
px = z
From the chain rule, expression (40) in the main text of this chapter,
i iiii Q
R
P
S
P
R
Q
SS
),(
R, P
Q
. … ..
],[ ],[
],[ ],[
),(),(
),(),(
p
Rpp
p
S
x
Rxp
p
S
p
Rpx
x
S
x
Rxx
x
S
PP
PP
),(
R, P
QS
],[],[ ),(),(
x
Rxp
p
S
p
Rpx
x
S PP
] [] [ xx
x
Rz
p
S
p
Rz
x
S
x
x
R
p
S
p
R
x
Sz
which gives,
),(x
),(R,R,
pP
xQSS z
The Poisson bracket is invariant under a canonical transformation (up to a constant value z.)
___________________________________________________________________ Appendix-3 How to generate canonical transformations
A way to generate restricted canonical transformations will result along the way of attempting to classify them. Toward this end, lets analyze the following expression,
ii P i
i
i
i Qpx - (1)
where the variables correspond to the transformation,
),,( ; ),,( tt pxpxQQ
PP .
It turns out, when (1) is written as a function of ),,( tpx
it reveals much about the
transformation.
Basically, the strategy consist in showing that, when the transformation is (restricted) canonical (that is, there exist a
H=H ),,( tpx
and a K=K )( , P
Q ),
expression (1) turns out to be an exact total derivative of a scalar function F=F ),,( tpx
.
ii P i
i
i
i Qpx - =
expanding the second term
= )( PPP iikiit
Q
p
Q
x
Q i
k k
i
i k
k
k
i
i
i pxpx
-
,
ik
andmi
Renanimng
,
ik
andmi
Renanimng
)Pt
QP
p
QP
x
Q -
i
ii
i
mi
i
m
m
m
m i
i
i
mi
i
i pxpx
i
i
m i
m
ii m
i
i
mi
t
Q
p
Q
x
Qpxp iimm P)P()P( (2)
i i
The objective is to show that there exists a function F that makes (2) equal to
t
Fp
p
Fx
x
Fi
i
i
i
ii
m i
mi
x
Qp mi P implies
m ki
m
m ik
m
m i
m
kk
i
xx
Q
xx
Q
x
Q
xx
mm
2
m
P)P()P(
m k
mk
x
Qp mk P implies
m ik
m
m ki
m
m k
m
ii
k
xx
Q
xx
Q
x
Q
xx
mm
2
m
P)P()P(
which gives,
PQ
,mm },{PP
ki
m i
m
km ik
m
i
k
k
i xxx
Q
xxx
Q
xx
the latter is the Lagrange bracket
Without proof we state:
The transformation ),,( ; ),,( tt pxpxQQ
PP is canonical
if and only if (3)
ikkikiki pxppxx PQPQPQ
,,, },{ and 0},{},{
Therefore, one obtains
0
i
k
k
i
xx
This indicates that there exist a function F such that,
x
F
(4)
Similarly
m i
mi
p
QmP implies,
m ki
m
m ik
m
k
i
pp
Q
pp
Q
p
mm
2 P)()P(
m k
mk
p
QmP implies,
m ik
m
m ki
m
i
k
pp
Q
pp
Q
p
mm
2 P)()P(
which gives,
PQ
,mm },{)P
)P
)( ki
m ki
m
ik
m
i
k
k
i pppp
Q
pp
Q
pp
= 0
This indicates that there exist a function F such that,
p
F
(5)
i
i
t
QΨ iP implies,
)P( i
2
k
ii
i k
i
k x
P
t
Q
tx
Q
x
Ψ
and )P( i
2
k
ii
i k
i
k p
P
t
Q
tp
Q
p
Ψ
m k
mk
x
Qp mk P implies
m k
mk
p
QmP implies
)P( m
2
t
P
x
Q
xt
Q
t
m
k
m
m k
mk
and )P( m
2
t
P
p
Q
pt
Q
t
m
k
m
m k
mk
This gives,
)(t
P
x
Q
x
P
t
Q
xt
i
k
i
i k
ii
k
k Ψ
and )(
t
P
p
Q
p
P
t
Q
pt
i
k
i
i k
ii
k
k Ψ
(6)
On one hand iQ can be determined by the Hamiltonian K=K )( , P
Q
i
iP
KQ
On the other hand )(tQi is a quantity that can also be tracked by the
Hamiltonian H=H ),,( tpx
t
QHQQ i
ii
),(],[ p
x
Thus
) (
],[],[ ),(),(
x
H
p
Q
p
H
x
Q
P
K
HQP
KHQQ
t
Q
ii
i
i
i
iii pp
xx
Similarly,
) (
],[],[ ),(),(
x
H
p
P
p
H
x
P
Q
K
HPQ
KHPP
t
P
ii
i
i
i
iii pp
xx
Further,
k
iii
k
i
ik
ii
x
P
x
H
p
Q
p
H
x
Q
x
P
P
K
x
P
t
Q
) (
) (x
H
p
P
p
H
x
P
x
Q
Q
K
x
Q
t
P
x
Q ii
k
i
ik
ii
k
i
Upon subtraction
k
iiiii
k
i
ik
i
k
i
i
i
k
i
k
ii
x
P
x
H
p
Q
p
H
x
Q
x
H
p
P
p
H
x
P
x
Q
Q
K
x
Q
x
P
P
K
t
P
x
Q
x
P
t
Q
) () (
) () (
x
H
x
P
p
Q
p
H
x
P
x
Q
x
H
x
Q
p
P
p
H
x
Q
x
P
Q
K
x
Q
x
P
P
K
t
P
x
Q
x
P
t
Q
k
ii
k
ii
k
ii
k
ii
ik
i
k
i
i
i
k
i
k
ii
x
H
x
Q
p
P
x
P
p
Q
p
H
x
P
x
Q
x
Q
x
P
Q
K
x
Q
x
P
P
K
t
P
x
Q
x
P
t
Q
k
ii
k
ii
k
ii
k
ii
ik
i
k
i
i
i
k
i
k
ii
) () (
Now, summing on the i variable (see expression (6)),
x
H
x
Q
p
P
x
P
p
Q
p
H
x
P
x
Q
x
Q
x
P
Q
K
x
Q
x
P
P
K
t
P
x
Q
x
P
t
Q
k
ii
k
ii
ik
ii
k
ii
ik
i
k
i
ii
i
k
i
k
ii
) () (
i
i
k
k
k
kk
ki
i
k
i
k
ii
x
HK
x
H
x
K
x
Hpx
p
Hxx
x
K
t
P
x
Q
x
P
t
Q
)(
0
},{},{ ,,
PQPQ
kk
k
x
HK
xt
Ψ
)(
(7)
But, according to (4), k
kx
F
kkk x
HK
xx
F
t
Ψ
)(
t
F
xx
HK
x kkk
Ψ
)(
0
) (0t
FHK
xΨ
k
(8)
Similarly, we expect to obtain
) (0t
FHK
pΨ
k
(9)
This implies that ) (t
FHKΨ
is a function of t alone. However, (4) and (5)
determine F up to an additive function f(t); the latter can be choose such that,
t
FHKΨ
(10)
From (2), which defines , and (10)
t
F
t
QHK
i
i
iP
From (2), which defines i , and (4)
im i
m
x
F
x
Qp
mi P
From (2), which defines i , and (5)
im i
m
p
F
p
Q
mP
___________________________________________________________________ 1 H. Goldstein, Classical Mechanics, Addison-Wesley Publishing (1959).
2 Saletan and Cromer, “Theoretical Mechanics” Wiley, 1971.), page 180.
3 Ref.2, page188.
