2 Classical Mechanics

7/23/2019 2 Classical Mechanics

1/26

Operators ( ) ( )Of x g x=

( )23 6d

x xdx

=

( ) ( )Of x of x=

( ) ( )O =f r g r

) ) ))

d d d

dx dy dz = + +

f(r f(r f(rf(r i j k

( ) ( ) ( )2

2

2

sin( ) sin( )d

n x n n xd x

=

- are performed on functions

-are performed on vector functions and

have directional qualities as well. Theseare referred to as vector operators.

-can obey the Eigen equation, and thus

have eigen values and eigen functions.

- In general we are concerned with the

function that obey this equation.


2/26

ClassicalMechanics-Position

( )x t

3 2( ) 2 1x t t t= +

Example

( ) ( ) ( ) ( )t x t y t z t = + +r i j k

2( ) ( 1) 0t t t= + + +r i j k

Example

Notice that we are using a function of timeto describe the positionnot some

fixed value.

This function tells you the position at any point in time.


3/26

ClassicalMechanics-Position-3D

[ ] ( ) ( ) ( )x t t x t x= = = r i r i

[ ] ( ) ( ) ( )y t t y t y= = = r j r j

[ ] ( ) ( ) ( )z t t z t z= = = r k r k

( )[ ]

( ) ( ) ( )

( ) [ ( )]

( )

( ) ( ) ( )

( ) ( ) ( )

t t

x y z t

t t t

x t y t z t

=

= + +

= + +

= + +

r r r

i j k r

i r i j r j k r k

i j k

Note that the operatoris applied to the position functionand the result is thequantity associated with the operator.

Ie. The xoperator give you the x component of r(t, this is !now as a

projection operator.

The vector operator rcan be constructed from the pro"ector operators.


4/26

ClassicalMechanics-Position-3D

2

( ) ( 1) 0t t t= + + +r i j k

[ ] 2 2 ( ) ( 1) 0 ( 1) ( )x t t t t x t = + + + = + = r i i j k

Example

[ ] 2 ( ) ( 1) 0 ( )y t t t t y t = + + + = =

r j i j k

[ ] 2 ( ) ( 1) 0 0 ( )z t t t z t = + + + = = r k i j k

( )[ ] [ ] [ ]

2

2

( ) [ ( )] ( 1) 0

( ) ( ) ( )

( 1) 0

t t x y z t t

x t y t z t

t t

= = + + + + + = + +

= + + +

r r r i j k i j k

r i r j r k

i j k


5/26

ClassicalMechanics-Velocity-D

( )( ) dx tv tdt

=

d

v dt =

[ ] [ ]3 2

2

( ) ( ) ( )

2 1

6 2

dv t v x t x t

dtd

t tdt

t t

= = = +

=

[ ] [ ]( ) ( ) ( )d

v t v x t x t dt= =

Example


6/26

ClassicalMechanics-Velocity-3D

( ) ( ) ( )( )

dx t dy t dz t t

dt dt dt = + +v i j k

d

v

dt

=[ ]( ) ( ) ( )dt v t t dt

= = v r r

[ ] 2

2

( ) ( ) ( 1) 0

( 1) 0

(2 1 0 )

dt v t t t

dt

d d dt t

dt dt dt

t

= = + + +

= + + +

= + +

v r i j k

i j k

i j k

Example


7/26


( ) ( ) ( )( ) [ ( )] ( ) ( ) ( )

x y z

dx t dy t dz t t t v t v t v t

dt dt dt = = + + = + +v v r i j k i j k

[ ] [ ] ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

x

x y z

x y z

v t x v t

d dx t dy t dz t x t x

dt dt dt dt

x v t v t v t

v t v t v t

=

= = + + = + + = + +

r r

r i j k

i j k

i i j k

( )x x

dv t v

dt= = i

y

dv

dt

= j z

dv

dt

= k


8/26


( )[ ] ( ) [ ( )] ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

x y z

x y z

t t v v v t

d t d t d t

dt dt dt

dx t dy t dz t

dt dt dt

v t v t v t

= = + +

= + +

= + +

= + +

v v r i j k r

r r r

i i j j k k

i j k

i j k

x y zv v v= + +v i j k

The corresponding vector operatorto velocity can be reconstructed from the

pro"ector operators of the components#


9/26


Example

[ ] 2 ( ) ( 1) 0 (2 1 0 ) 2 ( )x xd

v t t t t t v t dt

= + + + = + + = = r i i j k i i j k

[ ] 2 ( ) ( 1) 0 (2 1 0 ) 1 ( )y yd

v t t t t v t

dt

= + + + = + + = = r j i j k j i j k

2( ) ( 1) 0t t t= + + +r i j k

[ ] 2 ( ) ( 1) 0 (2 1 0 ) 0 ( )z zd

v t t t t v t dt

= + + + = + + = = r k i j k k i j k

( )

2 ( ) [ ( )] ( 1) 0

[ ( )] [ ( )] [ ( )]

( ) ( ) ( )

2 0

x y z

x y z

x y z

t t v v v t t

v t v t v t

v t v t v t

t t

= = + + + + + = + +

= + +

= + +

v v r i j k i j k

r i r j r k

i j k

i j k


10/26

ClassicalMechanics-!cceleration-D

2

2

( )( )

d x ta t dt=

2

2

d

adt

=

[ ] [ ]

2

2

23 2

2

( ) ( ) ( )

2 1

12 2

d

a t a x t x t dt

dt t

dt

t

= =

= +

=

[ ] [ ]2

2( ) ( ) ( )

da t a x t x t

dt

= =

Example


11/26

ClassicalMechanics-!cceleration-3D

2 2 2

2 2 2( ) ( ) ( )( ) d x t d y t d y t t

dt dt dt = + +a i j k

2

2

d

adt=

[ ]2

2

( ) ( ) ( )d

t a t t dt

= = a r r

[ ]

22

2 ( ) ( 1) 0

(2 0 0 ) ( )

d

a t t t dt

t = + + +

= + + =r i j k

i j k a

Example


12/26

ClassicalMechanics-"orce-D

2

2

( )( )

d x tF t m dt=

2

2

d

F m dt=

[ ] [ ]

2

2

23 2

2

( ) ( ) ( )

2 1

12 2

d

F t F x t m x tdt

dm t t

dt

mt m

= =

= +

=

[ ] [ ]2

2

( ) ( ) ( )d

F t F x t m x tdt

= =

Example


13/26

ClassicalMechanics-"orce-3D2 2 2

2 2 2

( ) ( ) ( )( )

d x t d y t d y t t m

dt dt dt

= + +

F i j k

2

2 dF m

dt=

[ ]2

2( ) ( ) ( )

dt F t t

dt= =F r r

[ ]

2

22( ) ( ) ( 1) 0

2 0 0

dt F t m t t dt

m

= = + + +

= + +

F r i j k

i j k

Example


14/26

#mpulse an$ Momentum

2

2

( )

( ) ( )

d x t

F t m ma td t= =

[ ] [ ] [ ]

( ) ( ) ( )p x t F x t dt mv x t= =

2

2

( )( ) ( )

d x tF t dt m dt mdv t

d t= = Momentum#mpulse

2

2

( )( )

d x tF t dt F t m dt m v F t m v

d t= = =

( ) ( ) ( )p t F t dt mv t= =#n general

"or a constant force


15/26

Momentum-D

( )( ) ( )

dx tp t mv t m

dt= =

dp mv mdt

= =

[ ] [ ] ( )

( ) ( ) ( ) dx t

p t p x t mv x t mdt

= = =

[ ] [ ]3 2

2

( )

( ) ( ) ( )

2 1

6 2

dx t

p t p x t mv x t m dt

dm t t

dt

mt mt

= = =

= +

=

Example


16/26

Momentum-3D

( ) ( ) ( ) ( )( ) ( )

d t dx t dy t dz t t m t m m m mdt dt dt dt = = = + +

rp v i j k

d

p mv m dt= =[ ] [ ] ( )

( ) ( ) ( )

d t

t p t mv t m dt= = = r

p r r

[ ] [ ]

2

( ) ( ) ( )

( 1) 0

2 0

t p t mv t

dm t t

dt

mt m

= =

= + + +

= + +

p r r

i j k

i j k

Example


17/26

#mpulse-D

( ) ( )p t F t dt= ( )

( )

dV t

F t dx=

d

p Fdt Vdtdx

= =

( )( )

dV tp t dt

dx

=

( ) ( ( )) ( ( ))

( ( ))

p t p x t F x t dt

d V x t dt

dx

= =

=

$orce can be thought of as

a change in potential energywith change in position


18/26

#mpulse-D

[ ] [ ]2

3 2

2

3 2

2

( ) ( ) ( ) 2 1

2 1

6 2

dp t p x t F x t dt m t t dt

dt

d

m t tdt

mt mt

= = = +

= + =

[ ] ( )2

00

( ) 2 ( ) 2 ( ) / 2 cos( ) sin( )

( )

d kx t kxp x t dt kx t dt kx t dt t

dx t

= = = =

Examples

2

0

1 ( ( ))

( ( )) ( ) ( ( )) ( ) cos( )2 ( )

dV x t

V x t kx t F x t where x t x t dx t = = =

i In terms of the "orce operator#

ii In terms of the Potential operator#


19/26

#mpulse-3D

( ) ( )t t dt = p F ( )( )t V t= F

V dt= p

( ) ( )t V t dt =

p

( ) ( ( )) ( ( )) ( ( ))t t t dt V t dt = = = p p r F r r


20/26

!ngular Momentum

( ) ( )t t= L r p

[ ]( ) ( ) ( ) ( ) ( ) ( )x y zx t y t z t p t p t p t = L

( ) ( ) ( )

( ) ( ) ( )x y z

x t y t z t

p x p t p t=

i j k

L


21/26

!ngular Momentum

x y zL L L= + +L i j k

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x z y

y x z

z x y

L y t p t z t p t

L z t p t x t p t

L y t p t x t p t

=

=

=


22/26

!ngular Momentum

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

x z y

y x z

z y x

L t y t p t z t p t

L t z t p t x t p t

L t x t p t y t p t

=

=

=

x z y

y x z

z y x

dz dyL yp zp m y z

dt dt

dx dz

L zp xp m z xdt dt

dy dxL xp yp m x y

dt dt

= =

= = = =

x y zL L L= + +L i j k


23/26

%inetic Energy22 2( ) ( ) ( )

( )2 2 2

mv t m dx t p t K t

dt m

= = =

( ) ( ) ( ) ( ) ( ) ( )( )

2 2 2

m d t d t m t t t t K t

dt dt m

= = =

r r v v p p

2 22

2 2

m d pK mv

dt m = = =

2 2 2

m d d mK

dt dt m

= = =

r r v v p p

[ ]22 2 [ ( )] ( ) [ ( )] ( )

2 2 2

mv x t m dx t p x t K x t

dt m

= = =


24/26

Potential Energy

( ) ( )V d= r F r r( ) ( )V t F t dx=

( )( )

dV tF t

dx= ( ) ( )V

= F r r

&oo's (a)

( ) ( )F t kx t=

2( ) ( )

2

kV t x t =

Coulom*s (a)2

2( )

4 ( )

zeF t

x t=

2

( )

4 ( )

zeV t

x t

=

( ( )) ( ( )) ( )V x t F x t dx t =

V Fdx=

( ( )) ( ( ))V t t d =

r F r r

V d= F r


25/26

Conservation of Energy

( ) ( )E K t V t= +Total energy remains constant,

as long as % is not an explicitfunction of time. (i.e %(x(t

[ ] [ ] [ ] ( ) ( ) ( ) 0dE d d

K x t V x t K V x t

dt dt dt

= + = + =

[ ] [ ] [ ] ( ) ( ) ( ) 0dE d d

K t V t K V tdt dt dt

= + = + = r r r

2 ( ) ( )2 2

pH K V V x V

m m

= + = + = +

p pr

&amiltonian


26/26

Conservation of Energy-&oo'+s (a)

[ ] ( ) 0 ?dE d

K V x tdt dt

= + = 2( ) ( )

2

kfor V x x t=

[ ] [ ]2 2

( ) ( )

2 2

p x t k x tdE d

dt dt m

=

2

2

2

2

1 ( ) ( )2 ( )

2 2

( ) ( ) ( )2 ( )

2

d dx t k dx t m x t

m dt dt dt

m dx t d x t dx t kx t

dt dt dt

=

=

[ ] [ ] [ ]( )21 1

( ) ( ) ( )2 2

d dp x t p x t k x t

m dt dt =

( )

2

2

( ) ( )( )

( )( ) ( ) 0

dx t d x t m kx t

dt dt

dx tma t F t

dt

=

= =&ince# Newtons 'aw

$ ma ) *

[ ] [ ]1

( ) ( ) 2 ( ) ( )

2 2

d d d k d m x t m x t x t x t

m dt dt dt dt

=

2 Classical Mechanics

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