chapter 4 hamilton variational principle hamilton jacobi eq classical mechanics 1

8/17/2019 Chapter 4 Hamilton Variational Principle Hamilton Jacobi Eq Classical Mechanics 1

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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 classicalmotion 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 )


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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 bracket4.4C 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


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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, z 1), ( x2, y2, z 2), …, ( xN , yN , z N ).

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 )(t q 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,t 1) and (b, t 2) the classical action S isdefined as,


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S (q ) ≡ )(

)(

2

1

t b,

t a,

L( )(t q , )(t q , t ) dt Classical action (4)

Function

or

set of functions (n-dimension case)

Number (1-dimension case)

orset 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 t 1 and t 2. 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, t 1) to (b, t 2), the system takes onlyone.

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 q q = 0 The variational principle (5)

That is, out of all possible paths by which the system could travel from an initial position

at time t 1 to a final position at time t 2, 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.


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x(t )

t t 1 t 2

xC

a

b

Figure 1. For a fixed points (a, t 1) and (b, t 2), 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

t b,

t a,dt t , x x, L =

)(

)( 2

12

1

][ )(t b,

t a,dt xV xm 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, t 1) and (b, t 2.)

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 areinfinitesimally 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,


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h(t 1)= h(t 2) =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 notthe 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 t 1 t 2

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),

d α

dS

)(

)(

2

1

t b,

t a, {m

2

1

2 [ 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.


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0 d

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

t t h -

)(

)(

2

1

t b,

t a, x C (t ) h(t ) dt

According to (8), h(t ) vanishes at t 1 and t 2 ; 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,

0 d

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 , x L(x, = )(

)( 2

11

1][ )(

t b,

t a, dt xV xm2

…

.

takes an extreme value when the path x(t ) .

satisfies the Newton’s Law x (t ) = – dx

dV = F

(13)


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In the next section, a key expression we will use through the derivation is the following:

Consider an arbitrary function,

)( t , ,S S 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 , ,S dt t , ,S vuδvΔu +

+ 33

2

2

1

1 uuu

S S S

+

+ 33

2

2

1

1 vvv

S S S

+ dt t

S

Sometimes the following notation is used,

)u

,u

,u

( 321

S S S S u

Accordingly,

)( )( t , ,S dt t , ,S vuδvΔu +v

δ

u

Δ

S S + dt

t

S

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 ), 1 x (t ), 2 x (t ), ,…, n x (t ) )the action is defined as

S( x ) ≡

)(

)(

2

1

t ,

t ,

b

a

L( 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 (t 1) = h (t 2) = 0


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x (t )= ( x1(t ), x2(t ), … , xn(t ), 1 x (t ), 2 x (t ), ,…, n x (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

t L( x ((t ), x ((t ), t ) dt (16)

From (15) and (16) one obtains,

d

dS

2

1

t

t { i

ii

i

ii

h x

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

h x

L

} dt +

2

1

t

t

i

ii

h x

L

-

2

1

t

t { i

ii

h x

L

dt

d )(

} dt

Since h

(t 1) = h

(t 2) = 0, one obtains

d

dS

2

1

t

t { i

ii

h x

L

} dt -

2

1

t

t { i

ii

h x

L

dt

d )(

} dt

= i

2

1

t

t {

i x L

- )(

i x L

dt d

} ih dt

In the last expression i x

L

and )(

i x

L

dt

d

are evaluated at x

(t ) + )(t h

.

0 d

dS

i

2

1

t

t {

i x

L

- )(

i x

L

dt

d

} )(t hi dt

In the last expression

i x

L

and )(

i x

L

dt

d

are evaluated at x

(t ).

The variational principle requires that,

0 d

dS

i

2

1

t

t {

i x

L

- )(

i x

L

dt

d

} )(t hi dt = 0 (17)

for arbitrary trial functions )(t hi .


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This could be satisfied only if,

0 x

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 i x are specified. That is, the xi and i x

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 , ,S S vu ,

3

3

2

2

1

1

uu

uu

uu

S S S

dt

dS

+ 33

22

11

vvvvvv

S S S +

+t

S

Sometimes, the following notation is used )u

,u

,u

( 321

S S S S u

Accordingly,

dt

dS

vv

uu

S S

+t

S

Example: The Lagrangian for a particle in an electromagnetic field

This is a case where the force depend on the velocity,

)v( B E F q (19)


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Since the Maxwell’s equation states that 0 B , then B can be expressed in terms ofa vector potential )( t ,x AA

AB (20)

(because the divergence of the rotational is identically equal to zero.)

On the other hand, the third Maxwell’s equation 0/ t B E can then be written

as 0)(

t

AE , or 0)/( t AE . 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( B E x qm

can be expressed as,

t qqm

A

x + )( Ax q (22)

It can be demonstrated (see homework assignment) that,

)( Ax )(-

A x

x A

x (23)

Accordingly, the vector expression (22) can be expressed in term of the components,

t

A

x x

q

m

+ )(

A x

x A

x = 1, 2, 3

The second and the fourth terms on the right side constitute a

exact differential of A

x

+dt

dA

x

A

x (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 )()(

2x x qm1/2 L . The magnetic

field introduces a term depending on the velocity. Hence, let’s try,

)()(2

2

1t , , f qm L x x x x (25)

where ),,( 322 x x xx


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x

f

xq

x

L

x

f

xm x

L

x

f

t x

f

x

f xm

x

L

dt

d

)()(x

x x

x

where x

stands for ),,(321 x x x

x

stands for ),,(321 x x x

0

x

L

x

L

dt

d

implies

x

f

t x

f

x

f

x

f

xq xm

)()(

x x

x x (26)

which should be compared with (24)

xm x

q

+

dt

dAq

xq

A

x

Notice the right side of (26) contains a term in x , which is not present in (24). Therefore

we require

x

f

)(

x x = 0

Or, equivalently,

x

f

x

for = 1, 2, 3 (27)

A particular solution of (27) is,

321 G xG xG xt , f 321 )( x G x (28)

Replacing (28) in (26),

Gt

G x x

q xm

)(

x x

G x

The last two terms on the right side constitute a

exact differential of G


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Gdt

d

x xq xm

G x (29)

which should be compared with (24),

xm

xq

+ dt dA

q xq

A

x

A comparison between (24) and (29) indicates that AG q . Replacing this solution in(28) gives,

)( t ,q f x Ax (30)

The Lagrangian in (25) is then given by,

)( )()(2

2

1t ,qqmt , , L x Ax x x x x (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 momentumThere 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


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L L(x , x ) (33)is,

dt

dL

i

[i x

L

dt

dxi +

i x

L

dt

xd i ]

From the Lagrange equationii x

L

dt

d

x

L

, the previous equation can be written as

dt

dL

i

[ (i x

L

dt

d

)dt

dxi +

i x

L

dt

xd i ]

i

[ (i x

L

dt

d

)dt

dxi +

i x

L

dt

xd i ]

i

( )(i

i x L x

dt d

dt

d

i

i

i x

L x

This implies,

i

i

i x

L x

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

i x

L x

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

1k k xm V (x ) (36)


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i x

L

= ii xm

i

i

i

x

L x

=

i

2

ii xm

= 2 T (x )

H ≡ i

i

i x

L x

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 mechanicsAn 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 whensolving 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, 1 x , 2 x , …, n x , t ) later specified beto

( x1, x2, … xn, p1, p2, … pn,, t )

and, instead of ),,( t L x x , 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 ),,( t x x and ),,( t p x variables is suitable for our purpose), let’s familiarize

with the following, much simpler, Legendre transformation


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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 d v (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 d v – v dy

using (33)

= u dx + v dy – y d v – v dy

dB = u dx – y d v (36)

(As we said above, we assumed that the definition v = y / y x, 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 d v (the new

independent variables,) which is what we wanted.

The last expression also gives,

u = x

B

and y = – v

B (37)

In summary,


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( 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 d v

(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

),,( t L L x x

, 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,

),,( ),,( t t p x x x

(39)

where ),,(

t x

L

i

i p x x

H L

),,( ),,( i t t L H p x x x - p x i

i (40)

xi has to be expressed in terms of ),,( t p x

) , ( ),,(),,( , t p x t t Li p x p x x x - i

i

The Hamiltonian function

On one hand,

),,( t H H p x implies,


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dH = dt t

H dp

p

H dx

x

H i

ii

i

ii

(41)

On the other hand, the expression for ),,( ),,( i t Lt p xi

i H x x - p x given in (40)

implies,

dH = dt t

L xd

x

Ldx

x

Ldp x xd p i

ii

i

iii

ii

i

ii

)(i x

L

dt d

i p

)( i pdt

d

i p

Thus, the first and fourth sum-terms cancel out

dH = dt t

Ldx pdp x ii

ii

ii

(42)

From (41) and (42) one obtains,

),,( ),,( i t t L H p x x x - p x i

i

) , ( ),,(),,( , t p x t t Li p x p x x x - i

i

Hamilton canonical

equations(2n first-order equations) (43)

i

i p

H x

i

i x

H p

i = 1, 2, …, n and

(44)t

L

t

H

Notice if L does not depend on t explicitly, neither does H .


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Recipe for solving problems in mechanics

i ) Set up the Lagrangian ),,( t L x x

.

ii ) Obtain the canonical momenta ),,(

t x

L

i

i p x x

iii ) Obtain ),,( ),,( i t Lt p xi

i H x x - p x

in terms of x

and p

.

Example. Two dimensional motion of a particle in a central potential V( r

) =V(r ).

L( ),,, r r )= T - V = 2

1m ( 222 r r ) - V(r ).

Using the definition (39),

r

L pr

= m r and

L p = m 2r

Using the definition (40),

),,,( ),,,(

r

r r Lr r p p p pr H -

p pr r

2

1m ( 222 r r ) V(r ).

The expressionsr

L pr

and

L p are inverted to

express r and in terms of ),,,( p pr

p

p p

p

mr m

2r r

2

1m ( 2

2

22r )()(mr

r m

p p )

),,,( p pr r H = 222

r

2 2 mr m

p p V(r ).

Using (30) one obtains the equation of motion for each of the 4 independent variables

p pr r ,,, .

r p

H r

=m

pr

p

H

= 2mr p

r

H pr

=

r

V

H

p 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:


20/64

t

H

dt

dH

(45)

Proof: dt dH

t

H p p

H x x

H i

ii

i

ii

- i p i x

The first and second sum-term cancel out, which proves the statement. That is,

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)( iii

xk T and L then has the form )( )( 2 x

V xk Li

ii

One obtains,i x

L

= ii xk 2 and

i

ii

i i

i xk x

L x

2)(2

= 2T

H =i

i

i x

L x

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). .


21/64

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

p x and coordinates.That is, the state of a system is specified by a point ),( p x

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.

(

,x )

t=t o

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 ,p x

and calculate its total

differential change with time,

dt

dF

t

F p

p

F x

x

F i

ii

i

ii

i p

H

i x

H

-

t

F

x

H

p

F

p

H

x

F

iiiiii


22/64

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

F H F, (49)

In terms of the Poisson bracket, the time dependence of a physical quantity is expressed

as,

dt

dF H F,

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, thatthere 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

1 x x x xt x x x x L (49)

ii xt x x x x x

L

i

p

),,,,( 2121 for i= 1,2 (50)

),,(),,( t L - xt ii

i p H x x p x

),,,,(),,,,( 212122112121 t x x x x L - x xt p p x x p p H

2211 p p x x )()(2

2

2

1

2

2

2

12

1

2

1 x x x x

where we have to replace the i x in terms of the i p given in (50)

2211 p p p p )()(2

2

2

1

2

2

2

12

1

2

1 x x p p

),,,,( 2121 t p p x x H )()(2

2

2

1

2

2

2

12

1

2

1 x x p p (51)

Without finding the explicit solution, let’s show that expression (48) tell us that the

angular momentum F,

12212121 ),,,,( p x p xt p pqq F angular momentum (52)


23/64

Is a constant of the motion.

First, let’s calculate,

1212

1111

x x p p x

H

p

F

p

H

x

F

2121

2222

x x p p x

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,

0dt

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 =i x

L

is a constant of motion. But the Hamiltonian

equationi

i x

H p

implies also that in this case 0

i x

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

t t L H p x x x - p x i

i

. Notice, H differ from – L by

i

p x

i 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.


24/64

xi i=1, …, n does not appear in the Hamiltonian

Under those conditions all the conjugate momentai

i x

H p

will be constant.

ii p

The Hamiltonian then may be written in the form

) , ... , ,( n21 H H

and the equation of motion for the coordinates x i will be,

)( constant H

xi

i i

whose solutions are,

i i t xi

It is true that rarely occurs in practice that all the coordinates are cyclic (for, thus, takingadvantage 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 ),( p x 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 ,p p x x .

) ( ),,,( ,, t p x L H t , i p - p p x x x i

i (55)

As a first step, let’s apply the variational principle to

),( p x S ) ≡

)(

)(

222

111

t ,

t , ,

, , p x

p x

),,,( t , L p p x x dt (56)

Inside the integral, we have just written ),,,( t ,p p x x for

simplicity, but ),, ,( )()()()( t ,t t t t p p x x should be used

instead, respectively.


25/64

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,

0 x

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), ) ( ),,,( ,, t p x L H t , i p - p p x x x i

i ,

i pi x

L

,ii x

H

x

L

-

0 x

L

x

L

dt

d

ii

impliesi x

H i p

(58a)

On the other hand,

0

i p

L

,

i

i

i p

H x

p

L

-

0 p

L

p

L

dt

d

ii

impliesi

i p

H x

(58b)

That is, we obtain in (58) the canonical Hamiltonian equations (43).

4.3 The Poisson bracketExpression (48) evaluates how a given dynamic quantity F= F ),( t ,p x

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 ),( p x

with the

dynamical variable R= R ),( p x

is defined as,

i iiii x

R

p

S

p

R

x

S RS, (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 ),( p x

. For, it may happen that an (invertible) transformation of

coordinates


26/64

),,( t p x Q Q

and ),,( t p x P P

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

F H F

],[ (58)

4.3A The Hamiltonian equations in terms of the Poisson brackets

In the particular case that F = x , expression (58) gives,

][][ H , xt

x H , 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,

],[],[

H pt

p H p

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

],[ H pdt

dp

i

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


27/64

i iiii x

x

p

x

p

x

x

x x x

, = 0 for any 1, 2, ..., n

i iiii x

p

p

p

p

p

x

p

p p

, = 0 for any 1, 2, ..., n

i iiii x

p

p

x

p

p

x

x p x

, = 0 for ≠

i iiii x

p

p

x

p

p

x

x p x

, = 1 for =

That is,

0, ,)()( p x, p x, ][][ p p x x (60)

,)( p x,][ p x … ..

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 p x , were calculated with

respect to other arbitrary variables (Q,P ), the value of )(, P Q,][ p x would be, in general,

different than .

(61)For an arbitrary transformation

( x,p) (Q,P )

, )( P Q,][ p x

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 ,)( P Q,][ p x , regardless of

the new canonical variables used to describe the motion. Hence the usefulness of thePoisson 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 ),( P Q

and R= R ),( P Q


28/64

where ),,( t p x Q Q

and ),,( t p x P P

i iiii

p x,

x

R

p

S

p

R

x

S RS )(, =

i

)( )( i

β

β i

β

β α i

α

αi

α

α p P

P R

pQ

Q R

x P

P S

xQ

QS

i

)( )( i

β

β i

β

β i

α

αi

α

α x

P

P

R

x

Q

Q

R

p

P

P

S

p

Q

Q

S

)

(

i

β

β i

α

αi

β

β i

α

α

i

β

β i

α

αi

β

β i

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

β i

α

αi

β

β i

α

α

i

β

β i

α

αi

β

β i

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

p x,

β α

α β

p x,

β α

α P

R P ,Q

Q

S

Q

RQ ,Q

Q

S )()(][][ (

+

) ][][)()(

β

p x,

β αα β

p x,

β αα P

R

P , P P

S

Q

R

Q , P P

S

Thus,

For S =S ),( P Q

and R= R ),( P Q

where ),,( t p x Q Q

and ),,( t p x P P

(62)

P

R P , P

P

S

Q

RQ , P

P

S

P

R P ,Q

Q

S

Q

RQ ,Q

Q

S

β

p x,

β α

α β

p x,

β α

α

β

p x,

β α

α β

p x,

β α

α β α

)()(

)()(

][][

][][

i iiii

p x,

x

R

p

S

p

R

x

S RS,

)(


29/64

Notice, if a particular transformation of coordinates ( x,p) (Q,P ) satisfies,)(],[ p x,α β P Q = ,

)(

],[ p x,

α β QQ = 0, and (63) )(],[ p x,α β P P = ,

then (62) gives )( p x, RS,

Q

R

P

S

P

R

Q

S

iiiii

. That is,

)( p x, RS, )( P Q, 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 remaininvariant, 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 ),,( t H p x

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

)( x p x (65)

22

2)(

)(21 x p2x-

x p

x p

That there is no H whose associated canonical equation ends up in (60) can be seen by

first requiring,

22)( x p p

H x

22

2)(

)(21 x p2x-

x p

x

x

H p

The first equation gives,

)(4)( 2222

x p x x p x

p x

H

and the second equation gives,

])()(2

1[ 222

2

x p2x- x p

x

p

x p

H


30/64

x p4x- x p

x - )(

)(21 2

22

The fact that

x p

H

p x

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

andQ

K - P

. The following transformation will do the trick.

xQ (67)22 )( x p P

The equations of motion for Q and P are,

xQ according to (65)

22 )( x p

= P

)2()( 2 x x p x p2 P

= 2 P 1/2

( p +2QQ )according to (65)

= 2 P 1/2

( 222

)()(2

1 x p2x- x p

x

+ 2QQ )

= 2 P 1/2

( P Q2 - P

Q

1/221 2QQ )

But P Q

= 2 P 1/2

( P

Q

1/221 )

Q P

In short, the motion described, in the x,p coordinates, as22 )( x p x

22

2)(

)(21 x p2x-

x p

x p

for which there is not a Hamiltonian,


31/64

is alternatively described in the new coordinates Q , P by the following equation of

motion,

P Q (68)

Q P

These equations do have a Hamiltonian. Indeed,

setting P P

K Q

, gives K (Q,P ) = (1/2) P 2 + 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) P

2(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 )(,)( t t p x

be the time development of a system on phase space.

This development is generated by a Hamiltonian ),,( t H p x

if and only if

every pair of dynamic variables ),,( t R p x

, ),,( t S p x

satisfies the

relation

],[ ][][dt

dS RS ,

dt

dRS R,

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 mathematicalburden in solving the Hamiltonian equations in the new coordinates; instead, more

often it is to identify more clearly those physical quantities that remain constants


32/64

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 whenmaking 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

)( ,)( t t p x (71)

for which a Hamiltonian exists; i .e. the changes of )(t x

and )(t p

are governed by a

Hamiltonian H :

i

i p

H x

and

i

i x

H p

(72)

Consider a transformation of coordinates,

),,( ),,( t t P

Q p x (73)

),,( ; ),,( t p x t p x Q Q

PP

As )(t x

and )(t p

change, the variables )(t Q

and )(t P

also change accordingly.

However, nothing ensures that the time evolution of the latter variables preserves the


33/64

Hamiltonian formalism given in expression (43). That is, it doesn’t always exists a

function K such thati

i P

K Q

andi

iQ

K P

.

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.

(

,x )

t=t o

t

phase space (

,Q )

t=t o

t

phase space

H

Transformation

Figure 3. For

),,( ; ),,( t p x t p x Q Q

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 ).,,( t p x

An invertible transformation of variables ),,(),,( t t P

Q p x such

that the time evolution of the new variables )(t iQ

and )(t P

(74)

preserve the form of the Hamiltonian equations,

i.e. there exists a function K such thati

i P

K Q

andi

iQ

K P

,

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


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Definition

A transformation that is canonoid with respect to all Hamiltonians (75)

is called canonical.

Canonical transformation theorem3

Let ),( p x

be a set of general coordinates on phase space. The Poisson brackets of anytwo dynamic variables R and S will be specified as,

i i i i i

p x

x

R

p

S

p

R

x

S S

),(R ,

Consider an invertible transformation

),,( ; ),,( t p x t p x Q Q PP

The following three statements are equivalent:

a) The transformation ),,( ;),,( t t p x p x Q Q PP is canonical. (76)

b) There exists a nonzero constant z such that any dynamical

variables R and S satisfy,

),(),( R ,R , p x Q S z S P (77)

(That is, the Poisson bracket is practically independent of the

coordinates used to calculate it.)

The transformation ),,( ;),,( t t p x p x Q Q PP is canonoid with

respect to all quadratic Hamiltonians of the form,

H= C +

n

pc xc1

)'(

+

+

n n

p p p x x x1 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.)


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4.4C Restricted canonical transformations

A canonical transformation ),,( ;),,( t t p x p x Q Q PP is called (80)

restricted-canonical transformation if in expression (78) z=1.

That is, ),(),( R , R , p x Q S S 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 , p x Q S S P . A way togenerate 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 Q p x - (81)

where the variables correspond to the transformation (73),

),,( ; ),,( t p x t p x Q Q PP (82)

It turns out, when (81) is written as a function of ),,( t p x

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 thefollowing alternative form,

ii P i

i

i

i Q p x - = else ...

t

F p

p

F x

x

F i

i

i

i

i i

(83)

To this end, let’s expand the left side in terms of the old coordinates,

ii P i

i

i

i Q p x - =

= ik P )(t

Q

p

Q

x

Q i

k

i

k

ii

i

i p x p x

k i k

k -

iQ

)PPP( imimi

i

i

i i

m

mm i

i

i

m

i

it

Q

p

Q

x

Q p x p x -

i

i

m i

m

ii m

i

i

mi

t

Q

p

Q

x

Q p x p iimm P)P()P( (84)


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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

K H t

F P

t

Q

i

ii

(85)

Further, replacing (85) in (84),

ii P i

i

i

i Q p x - =

t

F p

p

F x

x

F i

i

i

i

( i i

+ H - K )

Or

H - i

i

i p x -

K P Q i

i

i =

t

F p

p

F x

x

F i

i

i

i

i i

H - i

i

i p x -

K P Q 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 transformationsExpression (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 ),( p x 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:


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),(1 t , F Q x = ),( ),( t , F Q x p x

),(2 t , F Px = ),,( ),( t F Px p x ,

),(3 t , F Q p = ) ,( ),( t , F p x Q p

),(4 t , F Pp = ) ,( ),( t , F p x p P

Case 1: Assume the function ),(1 t , F Q q is given.

In this case, (86) states,

H - i

i

i p x -

K Q i

i

iP = ),(1 t ,

dt

dF Q x

=t

F Q

Q

F x

x

F i

i

i

i

111 i i

(87)

Since the xi and the Qi are independent, then the coefficients of i x and iQ should be

equal,

),(1 t , x

F

i

i p Q x

i= 1, 2, … , n (88a)

These are used to solve for the n variables

Qi as a function of ),( p x

),(1P t ,

Q

F

i

i Q x

i= 1, 2, … , n (88b)

Since the Qi have been determined in (88a),

this expression gives the n variables Pi as a

function of ),( p x

which leaves (87) with,

t

F H K

1 (88c)

Case 2: Assume the function ),(2 t , F Px is given.In this case, (86) states,

H - i

i

i p x -

K Q i

i

iP = ),(2 t ,

dt

dF Px


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=t

F F x

x

F i

i

i

i

222 P

P

i i

(89)

On the left side, we expand the termi

ii

QP

. For convenience we write it asm

mm

QP

,

H - i

i

i p x -

K Q m

m

mP = ),(2 t ,

dt

dF Px

=t

F F x

x

F i

i

i

i

222 P

P

i i

(89)’

where we will replace PP

mm ii

i

i i

m

Q x

x

QQ

i

m

m

mQ P = mm

i

i

i

i i

Q x

x

QP)P

P ( mm

i

Inverting the order of the summation

m

m

mQ P = PP

PP mm

i

im

ii

im

m i

Q x

x

Q

m

(90)

Replacing (90) in (89)’

H - i

i

i p x -

i

im

ii

im

m i

Q x

x

QP)

P ()( PP mm

m

+ K = ),(2 t ,dt

dF Px

=t

F F x

x

F i

i

i

i

222 P

P

i i

(91)

Since the xi and the Pi are independent in (91), then the coefficients of i x and iP

should be equal,

),( P 2m t , x

F

x

Q

i

m

m i

i p 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 ),( p x , since the Qm are involved there. Hence, we perform an extra step.

First, notice that the first equation in (92) can be written as,


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),( P 2m t , x

F Q

x im

mi

i p Px

)P( m2 m

mi

i Q F

x

p

(93a)

Also notice in the second equation of (92),

m

i

m

ii

m QQQ

m m m P

PP

P

P

)P( mmm

mm

i

QQ

m

im

m

PP

m

im

i

QQ

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

Q F (93b)

In term of the function F 2’,

m

)P( ' m22 mQ F F ,

expressions (92a) and (92b) give,

i

i x

F p ' 2

iQ =i

F

P

'2

(94)

K - H =t

F

' 2

At the end, there will be no interest in the function F 2. We already found a F 2’ that, if

expressed in terms of (x,P) as independent variables, it will generate a canonical

transformation. Hence, let’s just rename the F 2’ as F 2.

In summary

),( 2 t , x

F

i

i p Px

, iQ = ),(

P2 t ,

F

i

Px

, K - H =

t

F

2

solve for Once the Pi are known,


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this gives

),,(PP t ii p x ),,( t QQ ii p x

Invert them to obtain

),, t P x(Q x ),, t K P (Q = ),,( t H p x +t

F

'

2 ),( t ,Px

),, t P p(Q p

),, t K P (Q = ), ,( ),,),, t H t t P (Q P (Q p x +

+t

F

'

2 ),( ),, t ,t PP (Q x

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),( i xt , F P (95)

Let’s find out the canonical transformation it generates.

Using (94),

i

i

i x

F p P 2

(96)

iQ = ii

x

F

P

2

We find the new coordinates ) ,( PQ are the same old coordinates ),( p x . That is, the

function in i

x i2 ),( P xt , F iP generates the identity transformation.

),( p x I

) ,( PQ

ii) Infinitesimal transformations

Consider the generating function

i

x i2 ),( P xt , F iP + ),( t ,G p x (97)

where is an infinitesimal number, and ),( t ,G p x an arbitrary function (to be specified

later.)

Due to the small value of , and since i

i P xi generates the identity transformation, we

expect the new coordinates ) ,( PQ will differ from the old ones ),( p x also by

infinitesimal values; that is,


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iQ = i i x (98)

iP = i i p

Let’s figure out the values of i and i .

Applying (94) to (97) gives,

ii

i x

G

x

F p

i

2 P (99a)

iQ = ),(PP

)(2 t ,G x F

i

i

i

Pp x

based on (99a) one obtains

),( )( t ,G p

xi

i Pp x

i p iP

),( )( t ,G p

xi

i Pp x

(99b)

In short, we have obtained:

The function i

x i2 ),( P xt , F iP + ),( t ,G p x generates

the transformation

i

i x

G p

Pi (100)

iQ i

i p

G x

iii) The Hamiltonian as a generating function of canonical transformations

Notice in (100), if G is the Hamiltonian ),( t , H p x , and is chosen to be the incremental

time differential dt , one obtains,

i

i x

G p

Pi =

i

i x

H dt p

)(

Using the Hamiltonian equations (43)

= )()( ii pdt p

But )(dt pi is the increment of p due to the motion

= ii dp p

iQ i

i p

G x

=

i

i p

H dt x

)(

Using the Hamiltonian equations (43)

= )()( ii xdt x


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= ii dx x

That is, we have obtained:

The function i

x i2 ),( P xt , F iP + (dt ) ),( t , H p x generates the

transformation ),( p x ) ,( PQ

ii dx x Q i (101)

ii dp p Pi

where d xi and d pi 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

),( p x 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 ),( )()( t t ii p x 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 transformationSymbolically, the result in (101) can be expressed this way:

Defining )( t' t, F i

i P xi + (t ’ -t ) ),( t , H p x (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(t 0), p(t 0) (old coordinates) to the values of the state at

the time t , x (t ), p (t ) (the new coordinates).


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Canonicaltransformation

x (t 0), p (t 0) )( 0t t, 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

0 x

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 ableto 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 f t , F P)(),(2 x x P (106)

where the f i are arbitrary functions.

Using (94),

iQ = )(P

2 t f F

i

i

x

(107)

ii

i x

F p

2 =

m

m

i

m

x

t f P

)(x

That is, the new coordinates Qi depend only on the old coordinates and time,

chapter 4 hamilton variational principle hamilton jacobi eq classical mechanics 1

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