Scientific Computing Using Python - PHYS:4905 - Fall 2018
Lecture Notes #22- 11/14/2018 - Prof. Kaaret
These notes borrow from Linear Algebra
by Cherney, Denton, Thomas, and Waldron.
Masses on springs
Consider two masses, each with mass m, attached by springs
with spring constants as shown in the figure below. Let x1
be the displacement of the first mass from its equilibrium position
and x2 be the displacement of the second mass
from its equilibrium position.
What are the forces acting on each mass?
If we move mass 1, both springs that it is attach to will produce a
restoring force pushing it back towards it equilibrium
position. Therefore,
If we move mass 2, the spring that connects the two masses will
produce a force moving mass 1 away from its equilibrium
position. Therefore,
The total force on mass 1 is thus
We can do a similar analysis on mass 2 and we find that the total
force on mass 2 is
Equations of motion as linear operators
From Newton's laws, we know that
, or
Our force equations above have the force proportional to the
displacement. If we briefly consider the motion of mass 1 with
the position of mass 2 fixed, then the equation of motion for mass 1
is
Where I have been lazy and temporarily dropped the subscript 1.
The solution to this is a function of the form
where c is a complex number and we take the only real part
of the right hand side of the equation in order to keep things
real. Allowing complex values for c and taking the real part
is same as writing the function in the form
where A is the amplitude of the oscillation and t0
is the time of maximum displacement. We can relate these to c
as,
The math is easier in the complex form. In particular, the
derivatives are
Note that the latter looks like an eigenvector-eigenvalue
equation. If we think of
as a linear operator acting on the space of smooth functions of t,
then the eigenvectors of L are functions of the form
since L acting on
produces a function of the same form, multiplied by a
constant. In this content, the eigenvectors are called
eigenfunctions. What is the eigenvalue?
Substituting back into the equation of motion for mass 1 with
mass 2 held fixed, we find
So, we have a solution if
.
What about the second mass?
Going back to our problem with two masses, the equations of motion
are
Again, the solutions are of the form .
If we substitute those in, and then cancel the common time
dependence, we find
Do these look like matrix equations to you?
We can write
Then, our equations of motion become
which is an eigenvector-eigenvalue equation.
Let's solve the equation. (Play soothing music as students
solve the eigenvector-eigenvalue equation.)
First, we find the eigenvalues by solving
where we have set the eigenvalue
.
We find
What are the corresponding eigenvectors?
We substitute back into our matrix equation,
For the first eigenvalue, we get
so the eigenvector is
What does this mean physically?
Since we have
,
this means that
The masses move in the same direction and with the same
amplitude. This is the slower mode of oscillations, since this
is the lower frequency
.
For the other eigenvalue, we get the matrix equation
The solution is then
,
which means that the masses move in opposite directions (again with
the same amplitude). This is the faster mode of oscillations
as this is the higher frequency. We call these modes of
oscillation eigenmodes. Eigenvectors, eigenfunctions, and
eigenmodes are the same concept represented in different ways.
Looking at a physical system, we can excite the system into either
of the two eigenmodes by either pushing the mass in the same
direction (lower frequency) or towards each other (higher
frequency). See Coupled
oscillators.
Any arbitrary motion can be decomposed into the sum of the two
eigenmodes.
Change of basis
What is the vector space for our linear operator?
It depends on which linear operator you mean. If we go back to
our equations of motion,
We can rewrite this by introducing the linear operator
That acts on a vector consisting of two functions in time
representation the displacements of the two masses
The vector space for L is the space of all pairs of smooth
functions and
.
Our equation of motion is then the homogeneous equation
.
Most of the elements in this space are not solutions of the
equations of motion. In writing the matrix equation in the
previous section, we considered only solutions of the equations of
motion. This greatly reduces the vector space. The new
vector space, which is a subspace of the vector space of L,
is
where
This is the standard basis.
We can simplify this by transforming to the eigenvector basis.
The matrix M becomes diagonal in this basis. You should work
that out by constructing the change of basis matrix and doing the
change of basis. What are the eigenvalues?