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