small coupled oscillations. types of motion each multi-particle body has different types of degrees...
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Small Coupled Oscillations
Types of motion
• Each multi-particle body has different types of degrees of freedom: translational, rotational and oscillatory
Rf
CV 2
Formulation of the problem
• Let us consider a many-particle Lagrangian
• The system is stable, if each particle has a stable equilibrium position
• We assume small deviations from equilibrium ηi
6.1
),...,(),...,(2
11
1,1 n
n
jijinij qqVqqqqmL
iq0
iii qq 0
Formulation of the problem
• Kinetic energy of the system
• Potential energy of the system
6.1
n
jijinij qqqqmT
1,1 ),...,(
2
1
iii qq 0
n
jiji
kk
qqk
ijnij
nq
mqqm
1, ),...,(
001 ...),...,(2
1
001
...2
1
1,
n
jijiijT ...
2
~
ηTη
i
i
qqinn
nq
VqqVqqV
),...,(
0011
001
),...,(),...,(
...2
1
,),...,(
2
001
j
jii
qqjin
V
V
qi
Formulation of the problem
• We assume that this term does not vanish
• The Lagrangian of the system
• Equations of motion
6.1
...2
~
Vηη...
2
1
,),...,(
2
001
j
jii
qqjin
VV
V
qi
...2
1
1,
n
jijiijV
2
~
2
~VηηηTη
L 0L
n
jijiij
n
jijiij VTL
1,1,0 2
1
2
1
ii
LL
dt
d
00
0
11
n
jjij
n
jjij VT
Normal coordinates
• We have a system of linear ordinary differential equations of the 2nd order
• A natural choice of a trial solution
• Equations of motion result in
6.26.3
011
n
jjij
n
jjij VT 0VηηT
tik
n
kjkj
keCa
1
tik
n
kkjkj
keCia
1
2)( tik
n
kkjk
keCa
1
2
k
n
kjka
1
aζη
ζaλη 2kjkjk
VaζζTaλ VaTaλ VaaTaλa ** ~~
Normal coordinates
• Let us consider diagonal terms l = k
6.26.3
VaaTaλa ** ~~ ***** ~~ aVaλaTa *** ~~ VaaλTaa VV
TT
*
*
~~*** ~~ VaaλTaa aVaaTaλ
~~~~ *** VV
TT~
~
VaaTaaλ *** ~~
TaaλTaλa *** ~~
n
jijlij
*ki
*kk
n
jilljlij
*ki aTaλλaTa
1,1,
~~
0)(~1,
n
ji
*kklljlij
*ki λλaTa 0~)(
1,
n
jijlij
*ki
*kkll aTaλλ
0~)(1,
n
jijkij
*ki
*kkkk aTaλλ jkjkjk ia
n
jijkij
*ki aTa
1,
~
n
jijkjkijkiki iTi
1,
)()~~(
Normal coordinates
• If real α and β are assumed to be some velocities, then this expression has a form of a kinetic energy, which is always positively defined
• Thus if
6.26.3
n
jijkij
*ki aTa
1,
~
n
jijkjkijkiki iTi
1,
)()~~(
n
jijkijkijkijkijkijkijkijki TiTiTT
1,
)~~~~(
n
jijkijkiT
1,
~
n
jikjjiikT
1,
~
n
jikiijjkT
1,
~
n
jijkijkijkijki
n
jijkij
*ki TTaTa
1,1,
)~~(~ 0
0~)(1,
n
jijkij
*ki
*kkkk aTaλλ
*kkkk λλ
0)Im( kkλ
Normal coordinates
• Equations of motion do not have exponentially growing solutions
• This can be true only for two diagonal matrices
• We have a freedom of normalization for matrix a; let us impose the following normalization:
• Recall
• Then
6.26.3
0)Im( kkλ 0)Im( 2 k 0)Im( k tik
n
kjkj
keCa
1
TaaλTaλa *** ~~ 0)~()~( ** TaaλλTaa
ikiiik )~()~( ** TaaTaa
ikik )~( *Taa
VaaTaλa ** ~~
λVaa *~
Normal coordinates
• Equations of motion :
• We completely diagonalized our problem
• We have a generalized eigen-value problem
• Eigen-values of the problem are solutions of the secular equation:
• Eigen-vectors:
6.26.3
1Taa *~ λVaa *~
0VηηT
aζη
tik
n
kjkj
keCa
1
0VaζζTa 0~~ ** VaζaζTaa 0 λζζ
TaλVa
0
.........
......
...
...
312
31
222
22212
21
122
12112
11
TV
TVTV
TVTV
1Taa *~VaTaλ
TaVa λ
Normal coordinates
• Secular equation
• As the number of generalized coordinates increases, the power of the secular equation grows
• For very large systems, there are two ways to calculate eigen-values: analytical application of the group theory and computer calculations
• Modern applications: molecular vibrational spectroscopy, solid-state vibrational spectroscopy, etc.
6.26.3
0
.........
......
...
...
312
31
222
22212
21
122
12112
11
TV
TVTV
TVTV
Example: longitudinal oscillations of a CO2 molecule
• CO2 is a linear molecule; we will model it as follows:
• The Lagrangian
6.4
223
212
22
23
21 )(
2
1)(
2
1
2
1)(
2
1xxkxxkxMxxmL
Example: longitudinal oscillations of a CO2 molecule
• Secular equation:
6.4
223
212
22
23
21 )(
2
1)(
2
1
2
1)(
2
1xxkxxkxMxxmL
)222(2
))((2
13221
23
22
21
22
23
21 xxxxxxx
kxMxxm
0
332
33322
32312
31
232
23222
22212
21
132
13122
12112
11
TVTVTV
TVTVTV
TVTVTV
0
0
2
0
2
2
2
mkk
kMkk
kmk
Example: longitudinal oscillations of a CO2 molecule
• Eigen-vectors:
6.4
0
0
2
0
2
2
2
mkk
kMkk
kmk
0))2()(( 222 mMMmkmk
M
m
m
k
m
k 21;;0 321
1Taa *~VaTaλ
0)(
0)2(
0)(
32
2
322
1
212
jjj
jjjj
jjj
amkka
kaaMkka
kaamk
1
)(2
2
23
21
j
jj
Ma
aam
Example: longitudinal oscillations of a CO2 molecule
• Eigen-vectors:
6.4
Mm
mmMm
Mm
mMm
Mm
mmMm
212
1
2
1
2
1
212
20
2
1
212
1
2
1
2
1
a
)2()2(2
)(2
)(2
1
231
31
321
xxxMm
mM
xxm
mxMxmxMm
ζ
Example: longitudinal oscillations of a CO2 molecule
• Normal coordinates:
6.4
aζη ηaζ 1
Forced oscillations
• For open systems, we introduce generalized forces
• For each generalized coordinate, there is a component of a force
• We can introduce modified generalized forces for each normal coordinate
• Total work done
• Equations of motion:
6.5
ii F
j
n
jj
n
iii QF
11
ii Q
j
n
jiji a
1
j
n
jj
n
ij
n
jiji QaF
11 1
j
n
iiji QaF
1
Qλζζ
Forced oscillations
• Let us consider a periodic external force
• We look for a solution in the following form:
• After substitution into the equation of motion
• For generalized coordinates
• Resonance
6.5
j
n
jiji a
1
)cos(0 iii tQQ
Qλζζ
)cos()( iii tB
220)(
i
ii
QB
n
jj
j
jij tQa
122
0 )cos(
Questions?
Normal coordinates6.26.3
VaaTaλa ** ~~ ***** ~~ aVaλaTa *** ~~ VaaλTaa VV
TT
*
*
~~*** ~~ VaaλTaa aVaaTaλ
~~~~ *** VV
TT~
~
VaaTaaλ *** ~~
TaaλTaλa *** ~~ βαa i
Taa*~Taa*~
~*~~ aTa~
*~Taa
Taa*~ )()~~( βαTβα ii )
~~(~~ TαβTβαTββTαα i
*~Taa )()~~( βαTβα ii )
~~(~~ TαβTβαTββTαα i
)~~(
~~ TβαTαβTββTαα i
The independent coordinates of a rigid body
• Let us consider a many-particle Lagrangian
• The system is stable, if each particle has a stable equilibrium position
• We assume small deviations from equilibrium
6.1
),...,(),...,(2
11
1,1 n
n
jijinij qqVqqqqmL
iq0
iii qq 0
The independent coordinates of a rigid body
• Let us consider a many-particle Lagrangian
• The system is stable, if each particle has a stable equilibrium position
6.1
),...,(),...,(2
11
1,1 n
n
jijinij qqVqqqqmL
V
qi