In this video we are gonna study
secondary order systems of
linear differential equations and
normal modes. So here is a typical
system. Instead of giving you
the first derivative of x_1 and x_2
in terms of x_1 and x_2, the system
tells you the second derivative.
Since it's a second order differential
equation, the initial conditions have
to give you the initial values of x_1 and
x_2 and that initial derivatives of x_1
and x_2. This might come up from
physics where Newton's Law gives you
the acceleration but you have to tell
what the initial position and the initial
velocity are. It can also come up
in electrical engineering.
There are a lot of systems in the real
world described by equations of this
type and a nice thing about them
is that the matrices that you come up
with almost always are diagonalizable
with real eigenvalues.
So in this case, we will look at this 
matrix which is diagnolizable
with real eigenvalues.
So you write our initial conditions
in vector form. You write our equation
in this form. A dot means time derivatives
so two dots means second derivative
with respect to time.
Okay, our procedure is exactly the same
as with difference equations and
first order differential equations.
We diagonalize a matrix.
We pick new coordinates which are
the coordinates of X in a basis of
eigenvectors. We rewrite our equations
in terms of the y coordinates.
We solve one at a time and then we
convert the initial conditions for x
into initial conditions for y into final
values of y into final values of x.
So let's do things one step at a time.
Our matrix, 1 3 2 2. This is actually 
the identify plus the matrix we've
already seen before. The rows add
up to 4 so 4 has to be an eigenvalue.
And 1 1 has to be an eigenvector.
Since the trace is 3, the eigenvalues
add up to 4 so the other eigenvalue
has to be -1. And you can work out
that the eigenvector is 3 -2.
Great, we've done part one.
Next, we define y to be the coordinates
of x in the b-basis. And how is y related
to x? By change-of-basis matrix.
Here is a matrix whose columns
are b so this is P_EB. And this is its
inverse. So you take a value of x
it will give you a value of y. You take
a derivative of x and multiply by
P-inverse, that gives you the derivative
of y. Going the other direction,
you multiply by P rather than 
by P-inverse.
Now we write out our equations
in terms of y. Again, it follows
the same general pattern we've seen
before. Since x is y_1 b_1 + y_2 b_2,
the second derivative of x is the
second derivative of y_1 times b_1
plus the second derivative of y_2 times
b_2 because b_1 and b_2 are just fixed
vectors. They are 1 1 and 3 -2 for all
time. And Ax, well these are eigenvectors
so multiplying the first term by A
just gives you the eigenvalue λ_1.
The second term by A gives you
the eigenvalue of λ_2.
And the only way for these two things
to be equal to each other is that
y_1 double dot is λ_1 y_1 and y_2 
double dot is λ_2 y_2.
And in this case, the first eigenvalue
is 4. The second value is -1.
And we have our equations.
Equation for y_1 and the equation for
y_2 and they are decoupled.
And we solve the equations.
Equation for y_1, the y_1 double dot
is a positive number times y_1.
So we get cosh and sinh
and our kappa value is the square
root of 4 so that means 2. So we get
cosh(2t) and sinh(2t).
Initial value times cosh, plus the initial
derivative divided by kappa times
the sinh. The other equations have
our negative. y double dot is
- y_2. So the coefficient of -1
is less than 0 and our frequency
is the square root of, minus that
so the square root of 1, which is 1.
So we get cos (1t) and sin(1t)
initial value times cos
plus initial velocity, divided by
omega time the sine. Of course dividing
by 1 doesn't do very much.
And now we can convert. We take
our initial value and we multiply
by P-inverse and we get the initial
value for y.
We take the initial velocity for x.
We multiply by P-inverse,
we get the initial value of y-dot.
And then we can figure out what
y_1 is. See y_1 of 0 is 3. y one-dot
of zero is 1. So we get three times 
the cosh plus 1/2 times a sinh.
Initial value times the cosh plus
initial derivatives over 2 times
a sinh.
For y_2, initial value is 1.
Initial derivative is -2,
we get 1  cos(x) - 2  sin(x)
And once we have y_1 and y_2,
we can put them back together
to figure out x. x at any given time
is y_1 times the first eigenvector,
plus y_2 times the second eigenvector.
Or if you prefer, you could write this as
t * (y_1 y_2).
However you slice it, if you know y_1 and
y-2, you know x. We do know y_1 and
y_2, so there is x. So we wind up with 
some term that looks like cosh
of 2t in terms of looked like sinh of 2t.
Cosine. Sine.
The cosh and sinh of 2t have the same
value upstairs and downstairs
because they're being multiplied by
the vector 1 1. The cos and sin
come in the ratio of 3 parts upstairs
to -2 downstairs because they are
being multiplied by the eigenvector
3, -2.
So in other words, our whole picture
is we take our initial value and initial
derivative, use p-inverse to get initial
value and derivative of the y coordinates.
Solve these things one at a time.
Then convert back.
The only difference between this and
the first order difference equation
or differential equations is we could
write this horizontal step as
multiplication by a nice matrix. 
It was d to the n or e to dt.
Here is a little bit more subtle.
You have to solve the equations
and involves both initial values
and the initial derivatives.
But still, one variable at a time.
And that makes it easier.
Finally, normal modes.
When I say a mode, I mean
an eigenvector of A.
So in this case, the modes are
1 1 and 3 -2.
If the eigenvalue is positive,
then something that goes like
that eigenvector is gonna grow.
It's gonna have cosh and sinh
or grow for exponentials. 
It's gonna go as e to the kappa t,
where kappa is the square root of
the eigenvalue. In this example,
it was e to the 2t.
If λ is 0, you get constant plus constant
times t. We didn't have any in this
example, but you can. If λ is 0,
then you get oscillation.
Things go as sin and cos and
the frequency of oscillation,
actually it's the angular of frequencies,
number of radians per second,
is given by the square root of 
minus the eigenvalue.
And finally, here is a physical picture
what could be happening to give us
the differential equation we started
with. Imagine you have two blocks,
sitting on top of two hills. And there is
a spring connecting them and right now
the spring is relaxed. If you pull them
apart, the spring will pull them back
together. If you push them together,
the spring will push them apart.
One mode is x_1 is equal to x_2.
So that means that the two blocks
move in lock step. They both fall down
their hills. And they accelerate down
the hills because it's going downhill.
That's the mode 1 1. The mode 3 -2
has to do with their going apart.
And their tendency to fall down the hill
is more than overcome by the spring
pulling them back up and if you push
them in, the spring pushes them
back out. And the matrix isn't symmetric
because they weren't the same size block.
So this is the kind of system you can 
understand this motion.
You can understand this motion 
and an arbitrary motion is a linear
combination of something times
the first mode plus something times
the second mode.
