Undergraduate Classical Mechanics Spring 2017 Physics 319 Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 21
Undergraduate Classical Mechanics Spring 2017
Physics 319
Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 21
Undergraduate Classical Mechanics Spring 2017
Lagrangian Small Oscillation Theory
• Method for solving problems where several coupled
oscillations present
• Steps are
– Write Lagrangian for several oscillations including
coupling. If needed go into small oscillation limit
– Solve for system oscillation “normal mode” frequencies
– Solve for oscillation amplitude vector for each normal
mode
– Go into coordinates, the so-called normal mode
coordinates, where the oscillations de-couple, to solve
initial conditions and time dependences
Undergraduate Classical Mechanics Spring 2017
Two Masses and Three Springs
• Forces and equation of motion are
• Introduce 2 component “vector” describing state of system
1 1 1 1 1 2 2 1
2 2 2 2 2 1 3 2
F m x k x k x x
F m x k x x k x
1 1
2 2
x t x t
t tx t x t
x x
Undergraduate Classical Mechanics Spring 2017
Equations of Motion in Vector Form
• Equations of motion are
1 1 1 1 1 2 2 1
2 2 2 2 2 1 3 2
1 2 21
2 2 32
1 2 21
2 2 32
0
0
0
0
0
F m x k x k x x
F m x k x x k x
k k km
k k km
M K
k k kmM K
k k km
x = x
x x
Undergraduate Classical Mechanics Spring 2017
Sinusoidal Ansatz
• As we have done many times before assume sinusoidal
solutions of general form
• Simultaneous Linear Equations! Solution method from
Linear Algebra
Gives possible “normal mode” oscillation frequencies.
Then solve for associated (eigen)vector.
0
2
0
2
00 0
i t
i ix t x e
M M
M K M K
x = x
x x x
2det 0M K
Undergraduate Classical Mechanics Spring 2017
Case of Identical Masses and Springs
• Normal mode frequency problem an eigenvalue problem.
Solve normal mode (also called secular) equation
1 2 1 2 3
2
2
0
2 2 2
0 0
2 2 2
0 0
0 2det 0
0 2
/
2det 0
2
m m m k k k k
m k k
m k k
k m
2
2 2 4
0 0
2 2 2
0 0
2 0
2
Undergraduate Classical Mechanics Spring 2017
First Normal Mode
• Take minus sign solution
• Back in original matrix equation
• Such an oscillation in the system is the symmetric mode
• Masses move in the same direction with the middle spring
unextended. Oscillation frequency “obviously” satisfies
102
0 20 10
20
1 10
1 1
i t
xx x
x
At e
A
x
2 2
0 0
2 2
02 / 2k m
Undergraduate Classical Mechanics Spring 2017
Second Normal Mode
• Take plus sign solution
• Now normal mode eigenvector is
• Such an oscillation in the system is the antisymmetric
mode
• Masses move in the opposite directions with the middle
spring extended twice as much as the other two.
2 2
0 03 3
102
0 20 10
20
1 10
1 1
i t
xx x
x
Bt e
B
x
Undergraduate Classical Mechanics Spring 2017
In Pictures
Undergraduate Classical Mechanics Spring 2017
General Solution
• General solution for motion determined by 4 initial
conditions, giving the real and imaginary parts of A and B
• Picture of general motion
i t i tA B
t e eA B
x
2
2
2
0 2
2 2
0 2 0 2 0
/ , / 2 /
/ /
/ /
k m k m k m
k m k m
k m k m
Undergraduate Classical Mechanics Spring 2017
Normal Mode Coordinates
• General motion is simplified if go into coordinates tied to
the normal mode eigenvector pattern. Define
• These combinations will only oscillate at the normal mode
frequencies ω± separately, ξ1 at ω− and ξ2 at ω+
• By going into the normal mode coordinates, the coupled
oscillations problem becomes decoupled!
1 21
1 22
1 2
1/ 2
1/ 22
1/ 2
1/ 22
i t i t
x x
x x
t A e B e
x
Undergraduate Classical Mechanics Spring 2017
Case of Weak Coupling
• Expect slight frequency shifts in oscillators
• Normal mode eigenvectors are the same symmetric and
antisymmetric combinations that we saw before.
2 1 3
2 2 2
2 2
2
2 2
2
2
0det 0
0
/ / /
/ , / 2 /
k k k k
k k k m
k k k m
k m k m k m
k m k m k m
2
2
2
0 2
2 2
0 2 0 2 0
/ , / 2 /
/ /
/ /
k m k m k m
k m k m
k m k m
Undergraduate Classical Mechanics Spring 2017
General Solution
• Place following boundary conditions on solution
• Then get
• Phase delayed oscillations with amplitude that goes from
one degree of freedom to the other and back again
1
2
Rei t i t
x t A Ae e
x t A A
0
1 0 2 0
0
2 0 2 0
/ 2
cos cos /
sin sin /
A A x
x t t k t mx
x t t k t m
1 0 1 2 20 0 0 0 0 0x t x x t x t x t