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7/23/2019 2 Classical Mechanics
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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.
7/23/2019 2 Classical Mechanics
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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.
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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.
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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
7/23/2019 2 Classical Mechanics
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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
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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
7/23/2019 2 Classical Mechanics
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ClassicalMechanics-Velocity-3D
( ) ( ) ( )( ) [ ( )] ( ) ( ) ( )
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
7/23/2019 2 Classical Mechanics
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ClassicalMechanics-Velocity-3D
( )[ ] ( ) [ ( )] ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
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#
7/23/2019 2 Classical Mechanics
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ClassicalMechanics-Velocity-3D
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
7/23/2019 2 Classical Mechanics
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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
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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
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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
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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
7/23/2019 2 Classical Mechanics
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#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
7/23/2019 2 Classical Mechanics
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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
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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
7/23/2019 2 Classical Mechanics
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#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
7/23/2019 2 Classical Mechanics
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#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#
7/23/2019 2 Classical Mechanics
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#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
7/23/2019 2 Classical Mechanics
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!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
7/23/2019 2 Classical Mechanics
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!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
=
=
=
7/23/2019 2 Classical Mechanics
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!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
7/23/2019 2 Classical Mechanics
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%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
= = =
7/23/2019 2 Classical Mechanics
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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
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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
7/23/2019 2 Classical Mechanics
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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
=