Using Mathematica to solve

coupled oscillators

2 coupled oscillators between fixed walls (essentially the same

as coupled pendula)

Here we have two equal masses (m=1) and three springs (with spring-constants 1, c and 1). Thus c is

the strength of the "coupling" between the two masses, which otherwise oscillate independently.

This is the general solution with four integration constants, and a sum of two normal modes. (It's a little

hard to parse this as the result is given after complexificatin.)

DSolve@8x''@tD + H1 + cL x@tD - c y@tD � 0, y''@tD + H1 + cL y@tD - c x@tD � 0<,8x@tD, y@tD<, tD

::x@tD ®1

4

ã-ä t- -1-2 c t Jã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tN C@1D +1

4 -1 - 2 c

ã-ä t- -1-2 c t J-ã

ä t+ ä -1 - 2 c ã

-1-2 c t- ä -1 - 2 c ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tN

C@2D +1

4

ã-ä t- -1-2 c t J-ã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t- ã

ä t+2 -1-2 c tN C@3D +

ã-ä t- -1-2 c t Jãä t + ä -1 - 2 c ã -1-2 c t - ä -1 - 2 c ã2 ä t+ -1-2 c t - ãä t+2 -1-2 c tN C@4D

4 -1 - 2 c

,

y@tD ®1

4

ã-ä t- -1-2 c t J-ã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t- ã

ä t+2 -1-2 c tN C@1D +

ã-ä t- -1-2 c t Jãä t + ä -1 - 2 c ã -1-2 c t - ä -1 - 2 c ã2 ä t+ -1-2 c t - ãä t+2 -1-2 c tN C@2D

4 -1 - 2 c

+1

4

ã-ä t- -1-2 c t Jã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tN C@3D +

1

4 -1 - 2 c

ã-ä t- -1-2 c t

J-ãä t

+ ä -1 - 2 c ã-1-2 c t

- ä -1 - 2 c ã2 ä t+ -1-2 c t

+ ãä t+2 -1-2 c tN C@4D>>

We can pick out one normal mode by introducing initial conditions in which both x and y are moved by

the same amount to the side. This normal mode has omega=1.

The “Flatten” command strips away one of the parentheses:

sol1 = Flatten@DSolve@8x''@tD + H1 + cL x@tD - c y@tD � 0, y''@tD + H1 + cL y@tD - c x@tD � 0,

x@0D � 1, x'@0D � 0, y@0D � 1, y'@0D � 0<, 8x@tD, y@tD<, tDD

:x@tD ®1

2

ã-ä t I1 + ã

2 ä tM, y@tD ®1

2

ã-ä t I1 + ã

2 ä tM>

Here’s one way of getting the real part of x(t)

First pick out the x[t] part of the solution

Here’s one way of getting the real part of x(t)

First pick out the x[t] part of the solution

x@tD �. sol1

1

2

ã-ä t I1 + ã

2 ä tM

Expand to multiply out

Expand@x@tD �. sol1D

ã-ä t

2

+ãä t

2

Finally, ComplexExpand gives the answer in a simple form, which is real because the

initial conditions are real.

ComplexExpand@Expand@x@tD �. sol1DDCos@tD

ComplexExpand@Expand@y@tD �. sol1DDCos@tD

Here's the other normal mode, where x=-y, and omega=Sqrt[1+2c] (which agrees with our answer from

lectures when c=1):

sol2 = Flatten@DSolve@8x''@tD + H1 + cL x@tD - c y@tD � 0, y''@tD + H1 + cL y@tD - c x@tD � 0,

x@0D � 1, x'@0D � 0, y@0D � -1, y'@0D � 0<, 8x@tD, y@tD<, tDD

:x@tD ®1

2

ã- -1-2 c t J1 + ã

2 -1-2 c tN, y@tD ® -1

2

ã- -1-2 c t J1 + ã

2 -1-2 c tN>

Simplify@ComplexExpand@y@tD �. sol2D, c > 0D

-CosB 1 + 2 c tF

Here's a solution with boundary conditions that excite both modes:

sol = Flatten@DSolve@8x''@tD + H1 + cL x@tD - c y@tD � 0, y''@tD + H1 + cL y@tD - c x@tD � 0,

x@0D � 1, x'@0D � 0, y@0D � 0, y'@0D � 0<, 8x@tD, y@tD<, tDD

:x@tD ®1

4

ã-ä t- -1-2 c t Jã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tN,

y@tD ® -1

4

ã-ä t- -1-2 c t Jã

ä t- ã

-1-2 c t- ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tN>

Plot for coupling c=1 (the first case discussed in class):

2 CoupledOsc.nb

Plot@8x@tD �. sol �. c ® 1, y@tD �. sol �. c ® 1<,8t, 0, 30<, PlotStyle ® 88Thick, Red<, 8Thick, Blue<<D

5 10 15 20 25 30

-1.0

-0.5

0.5

1.0

Weak coupling (c=0.1): notice that x (red) mostly oscillates at first, then y (blue), etc.

This is like the demonstration in class where the energy oscillates back and forward between the two

pendula.

Plot@8x@tD �. sol �. c ® .1, y@tD �. sol �. c ® .1<,8t, 0, 120<, PlotStyle ® 88Thick, Red<, 8Thick, Blue<<D

20 40 60 80 100 120

-1.0

-0.5

0.5

1.0

Plot@8x@tD �. sol �. c ® .1, y@tD �. sol �. c ® .1<,8t, 0, 120<, PlotStyle ® 88Thick, Red<, 8Thick, Blue<<D

20 40 60 80 100 120

-1.0

-0.5

0.5

1.0

Very strong coupling (c=5). The two masses oscillate about an envolope corresponding to the low-

frequency normal mode.

CoupledOsc.nb 3

Very strong coupling (c=5). The two masses oscillate about an envolope corresponding to the low-

frequency normal mode.

Plot@8x@tD �. sol �. c ® 5, y@tD �. sol �. c ® 5<,8t, 0, 10<, PlotStyle ® 88Thick, Red<, 8Thick, Blue<<D

2 4 6 8 10

-1.0

-0.5

0.5

1.0

Making an animation (essentially equivalent to the coupled

pendula demonstration)

Convenient to pull out the functions x[t] and y[t] from solution:

xx@t_D = Re@x@tDD �. sol

1

4

ReBã-ä t- -1-2 c t Jã

ä t+ ã

-1-2 c t+ ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tNF

yy@t_D = Re@y@tDD �. sol

-1

4

ReBã-ä t- -1-2 c t Jã

ä t- ã

-1-2 c t- ã

2 ä t+ -1-2 c t+ ã

ä t+2 -1-2 c tNF

Place a disk at the positions of the masses, assuming (arbitrarily) that they are distance 4 apart at rest.

c0 is the coupling.

plotosc@t_, c0_D := Graphics@8Red, Disk@8N@4 + xx@tD �. c ® c0D, 0<, 0.1D, Blue, Disk@8N@yy@tD �. c ® c0D, 0<, 0.1D<,Axes ® True, PlotRange ® 88-1.1, 5.1<, 8-.14, .14<<D

Now make a little movie:

With weak coupling

Animate@plotosc@t, 0.1D, 8t, 0, 100<, AnimationRate ® 1D

t

-1 1 2 3 4 5

With strong coupling

4 CoupledOsc.nb

Animate@plotosc@t, 5D, 8t, 0, 100<, AnimationRate ® 1D

t

-1 1 2 3 4 5

Here’s the first normal mode (which is independent of the coupling c):

xx1@t_D = Re@x@tDD �. sol1 ; yy1@t_D = Re@y@tDD �. sol1;

plotosc1@t_D :=

Graphics@8Red, Disk@8N@4 + xx1@tDD, 0<, 0.1D, Blue, Disk@8N@yy1@tDD, 0<, 0.1D<,Axes ® True, PlotRange ® 88-1.1, 5.1<, 8-.14, .14<<D

Animate@plotosc1@tD, 8t, 0, 100<, AnimationRate ® 1D

t

-1 1 2 3 4 5

Here’s the second normal mode (which idepends on the coupling c, set to 1 below):

xx2@t_D = Re@x@tDD �. sol2; yy2@t_D = Re@y@tDD �. sol2;

plotosc2@t_, c0_D := Graphics@8Red,Disk@8N@4 + xx2@tD �. c ® c0D, 0<, 0.1D, Blue, Disk@8N@yy2@tD �. c ® c0D, 0<, 0.1D<,

Axes ® True, PlotRange ® 88-1.1, 5.1<, 8-.14, .14<<D

Animate@plotosc2@t, 1D, 8t, 0, 100<, AnimationRate ® 1D

t

-1 1 2 3 4 5

CoupledOsc.nb 5