Separability - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics

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Separability

Prinicipal

Function

•^

In

some

cases

Hamilton’s

principal

function

can

be

separated.

–^

Each

W

depends

on

only

one

coordinate.

–^

This

is

totally

separable.

Et

q W t qS

j k N k

k

j j^

^ 



(^

1

(^

1

t q W q W t

qS

j mj m j k m k

k

j j

^





^ 

Function can be partially separable.

Staeckel

Conditions

•^

Specific

conditions

exist

for

separability.

–^

H^

is^

conserved.

–^

L^ is

no

more

than

quadratic

in

dq

j / dt

,^ so

that

in

matrix

form:

H

( p

a ) T

p^ 

a )+

V (

jq )

–^

The

coordinates

are

orthogonal,

so

T^

is^

diagonal.

–^

The

vector

a^

has

a^ j

=^

a^ j^

j ( q

–^

The

potential

is

separable.

–^

There

exists

a^

matrix

with

= ij

( qij

i )

^

^

jj jj^

T

T^

1



j qj T jj V V^

^

^

^

jj j^

1 T

1 ^1

Combined

Potentials^ •^

Particle

under

two

forces

-^ Attractive

central

force

-^ Uniform

field

along

z

•^

Eg:

charged

particle

with

another

fixed

point

charge

in^

a

uniform

electric

field.

gz k r zr V^

2 2 2

z y x r^

 X

Y

Z

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Energy

and

Momentum

2 2

^

ds dt m T

2 2 2 2 2 2 2

^ 

d

d

d

dz dy dx ds^

^

^

^

m^4 dT d p

^

^

^

m^4 dT d p

^

^

m dT d p^

^

  ^

p m p p

m T^

2 2 2

^

g k

Substituting for the new variables: V

Separation

of

Variables

•^

Hamiltonian

is

not

directly

separable.

-^ Set

E = T + V

-^ Multiply

by



•^

There

are

parts

depending

just

on

•^

There

is

a^

cyclic

coordinate

-^ Constant

of

motion

p^ 

-^ Reduce

to

two

degrees

of

freedom

k E p m g p m E p m g p^ m









2 2 2 2 2 2