M1L1i.txt

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We have a rotating system.
We have a system which precesses.
So we want to learn about rotations in general,
and what I want to show you is that under very
general circumstances, we can solve
the equation, the dynamics of the system,
by going into rotating frame.
You all know about rotating wave tunnel, rotating wave
approximations.
That's in quantum physics.
I'm simply talking about a classical system,
and I want to solve the equations
for the classical system by going to rotating frame.
And I want to show you where this is exact and where not.
OK, so this is actually something
which we do in undergraduate, definitely in 8.012,
but let me remind you.
When we have a rotating vector, which
rotates with a constant angular momentum--
with a constant angular frequency,
then the time derivative of the vector is the cross product.
But now we want to allow-- so this is when the vector is
constant and just rotates-- but now we
want to assume that there is something else, as an arbitrary
time dependence of the vector in the rotating frame.
So we have a vector which changes according to A dot 0,
but it also rotates, and that means--
and this is exactly shown in classical mechanics--
that in the inertia frame, the time derivative
is the sum of the two.
It is the change of the vector in the rotating
frame plus omega cos of A.
So it has this equation.
Has the simple two limiting cases.
But if there is no change in the rotating frame,
then you retrieve the kinematics of pure rotation.

When our rotating frame is not rotating
or it rotates at zero angular frequency, then of course
the two time derivatives are the same.
But anyway what I derived for you is an operator equation
that the time derivative in the rotating frame
is related to the time derivative
in the inertia flame in this way.

And now we want to apply it to our angular momentum, L dot.

So this is just applying the operator equation
to our angular momentum, L. And now we
want to specialize that the equation of motion
for the angular momentum was that it's L cross gamma B.
And then I add this.
So if we now describe our precessing classical magnetic
moment, which has the equation of motion
that L dot is L cross gamma B. If we describe it in a rotating
frame, then the equation of motion
gets modified as follows.

Now what happens is we factor out the gamma.
Gamma L cross B is the real field.

So what we observe is that when we go into a rotating frame--
and this is exact-- that the real magnetic field gets
replaced by an effective magnetic field,
because there is an extra term added to it, which we can call
a fictitious magnetic field.

So this is just an exact transformation
of our equation of motion for a precessing
system into the rotating frame.
And now, of course, we haven't made any assumptions
what the rotating frequency is.
But if you would choose the rotating frequency
to be the Larmor frequency minus gamma times B,
then our effective magnetic field vanishes.
And then we know, because there is no magnetic field,
the angular momentum is constant in the rotating frame.

In other words, the dynamics of this system
means that L is constant in a rotating frame,
and if you want to know what happens
in the original, in the lab frame, in the inertia frame,