this program is brought to you by
Stanford University please visit us at
stanford.edu
now levels theorem is really at the
heart of quanta of classical mechanics
it's at the heart of Hamiltonian
classical mechanics and a quantum
mechanical version of it which is called
unitarity is very much at the heart of
quantum mechanics it will profit us to
understand levels theorem are because as
I said it is the immediate classical
analog of one of the most fundamental
principles of physics quantum mechanics
it can also be thought of as the idea or
generalization of the idea of
information conservation we talked about
this I think in the first lecture where
we just remind you I think I've done
this a couple of times but let me remind
you again that if you have a phase space
which consists of a collection of points
now by the phase space I simply mean the
space of possible configurations of a
system in the case of a mechanical
system it would be positions and
velocities in the case of just a system
with a finite number of states it would
be any one of those finite number of
states and then the laws of mechanics
for this very simple setup would simply
be expressed as a rule for going from
any configuration to any other
configuration and I can draw it in a
more complicated way but there's not
much point I could draw it sum of lines
cross but that doesn't add anything to
it you could have some disconnected
cycles but generally that's the pattern
and you'll notice that each point has
one arrow coming into it and one arrow
going out of it that's the basic idea of
information conservation
then nothing gets lost there's no let's
call it convergence or divergence of the
flow in the phase space this can be
thought of as a kind of flow ah what's
what's it what's the word for what they
do in discos where they flashlights at
you stroboscope stroke you could think
of it stroboscopic Li where each instant
are you flash and the system moves to
the next configuration so it's a sort of
stroboscopic flow of the points in the
phase space but um you could imagine
populating each of these points
populating means imagine a system at
that point in phase space undergoing the
motion that the system is supposed to
have so we would have a collection of
systems filling up the phase space one
at each point in the phase space and as
time evolves the points in the phase
space just move I have described this I
think probably the best way to describe
it is in that they move incompressible
among other things that means that if
you start with a separate point the
separate system at each point in the
phase space you'll never wind up with
two points or two systems at the same
point in the phase space it's not like
playing Monopoly where the monopoly men
can wind up on the same on the same
square that never happens they stay
distinct from each other
the points in the phase space they don't
diverge they don't converge what would
diverge mean diverge would be a
situation where an incoming line would
split in two now what would that mean
that would mean there was some ambiguity
in the evolution of the system if it
went from here to here it wouldn't know
where to go next
sometimes it might go this way sometimes
it might go that way that doesn't happen
so there's no divergence of the flow and
there's also no convergence of the flow
convergence would mean that from two
different phase points
you come to the same phase point then
you would always know where to go but
you wouldn't know where you came from to
find yourself over here you wouldn't
know if it came from here over here
those are the forbidden kinds of flows
as I said you can think of a lack of
convergence or divergence on the phase
space young it's quite thinkable it's
not that it's not thinkable to have
these things they don't occur in
classical physics they don't occur
except insofar as you ignore things
except insofar as you ignore degrees of
freedom as I've said over and over
friction is an example where you start a
system in any number of ways its are in
many ways it comes to the same
configuration well not quite but you get
the idea friction to slow things down
brings them to rest no matter how they
start they come to rest many many
different configurations can wind up
exactly the same but that's only because
you've ignored things you've ignored the
molecular structure of a tabletop you
can take this as an experimental fact or
you can take it as a consequence of the
basic rules of quantum mechanics but all
the classical mechanical systems that
are of interest in fundamental physics
have the property that there's no
divergence or convergence
yes it is saying something more than
that it is saying something more than
that and we'll see we'll see what it
says it's what Leo's theorem is what it
says but we'll come to it for example
the sort of thing that can't happen is
here's phase space okay every point in
this phase space is distinct from every
other point in some sense now you could
take a group of points and follow them
and they might all squeeze together into
a smaller volume all right that's the
kind of convergence that doesn't happen
instead the flow in phase space is a
kind of incompressible flow both
incompressible what's the opposite of
compression and non divergent well the
phase space doesn't blow up it doesn't
shrink but we will come to a precise
definition of it soon enough so in some
sense the phase space doesn't pinch
together or diverge it says though
here's the way you can think about it in
the continuum mechanics case you can
imagine populating the phase space let's
call it P and Q Q and P and of course
there may be many P's and Q's I simply
can't draw them on the blackboard and we
might imagine in the same way that we
populated each point here with a system
we can imagine populating the phase
space with a dust with a dust of points
and follow that dust of points let's
assume that the phase space is populated
uniformly with a constant density a
uniform density of points of dust and we
follow each point as it moves through
the phase space it defines a kind of
flow but it defines a special kind of
flow which is called incompressible and
it basically has the property that each
point in the phase space can be thought
of as having a little volume around
it each one occupies its own little
volume if we take the density to be
uniform then each one of these volumes
is the same and as we follow it when we
see it later each phase point is
separated from each other phase point by
the same amount of volume in the phase
space or you can say it another way if
you take a group of points there are
three ways to say it all of them
equivalent the first is to follow
individual points with little volumes
around them and watch them move okay ah
the volume of each little point or the
region occupied by that little little
point stays the same but we can
generalize that we can say take any
region of the subspace of the phase
space to begin with take any region and
follow its points follow its points with
time after time that phase space region
will change it'll change the shape some
points move faster than other points it
may spread one way it may stretch some
direction they may compress in another
direction but the one thing that's true
is that the volume of that region will
stay the same in other words the volume
of a fixed set of points in the phase
space will not change and it will be
maintain connectedness and it will
maintain connectedness absolutely they
don't disconnect the connectedness is
simply the predictability and it became
disconnected how could it become
disconnected a phase point would split
somehow which would simply mean that
didn't away at the goal the equations
broke down at this point and didn't tell
a phase point where to go properly so
yeah regions maintain their connectivity
in other words the region like this
can't break up into two disconnected
region
and the whole cannot appear so it's a
very continuous smooth the holes the
connectivity is conserved but the type
of connectivity simple kind of simply
connectedness multiple connectedness
whatever it is the topology of the chunk
of phase space is maintained as is its
volume what is not maintained in general
is the distance in the phase space
between points take a pair of points in
the phase space and follow them if we
follow them we may discover later that
they're closer together now you say how
is that consistent how is it possible
that the volume of a phase region
doesn't change but the distance between
points can change well what can happen
for example is a volume can stretch in
one direction and compress in the other
direction volume being conserved the
volume of all of this collection of
points in here is unchanged but if you
follow each point you'll find that the
shape changes but the shape changes in a
way that leaves the volume unchanged but
in particular this point and this point
are now found here and here so they're
much further from each other than they
were on the other hand this point and
this point are found here and here so
they're much closer together than they
were points switch plot no no everything
is extremely continuous
conserves orientation what does that
mean oh you mean two one well you can
certainly have a situation with is a
point in phase space moving to the right
and another point in the phase space
moving to the right and they start like
this and this one is moving faster than
this one and it passes it yes that can
happen no question that that that can
happen for example you can have a
situation where if you started with a
square in phase space that square could
evolve into a what does that call
parallel or a parallelogram of the same
area that would happen for example if
the top was moving faster than the
bottom but the width of the whole thing
that's a sort of sheer motion where the
top shears relative to the bottom that's
perfectly allowed that that conserves
the volume if the height and the width
stay the same conserves a volume so
that's possible and that would certainly
allow a point back here to get ahead of
a point over here yeah so uh well go
ahead ask the question yeah what's that
yeah the double pendulum in a
gravitation yeah absolutely
absolutely but but we'll yeah yeah we'll
talk about that case that is not a
counter example to this right it's um an
example of it but one which is less
smooth and less predictable but for
reasons we'll come to okay well let me
just say it very quickly
I say the phase space maintains a
certain degree of connectivity it
maintains its connectivity all it
happens as shape changes but the shape
can change wildly for example the sphere
can evolve into something with
I'm trying to keep the volume the same
as it was before but I'm probably not
horrible so the there's no rule that
says that the shape can't evolve into
some wildly extravagantly different
structure which eventually starts to
look more and more fractal now it never
really forms a fractal because
continuity tells you that things which
are close to each other will stay close
to each other but in time it'll start to
get more and more fragmented more and
more fractured but always maintaining
its volume okay maintaining its volume
so that distinguishable points remain
distinguishable and a measure of
distinguished ability is the volume
surrounding a point but we will return
to this particular picture I ought to
leave it there because I'll probably
wind up drawing it again but not right
now that incidentally is the basic case
of a chaotic system chaotic systems do
exactly this kind of thing non chaotic
systems tend to maintain a greater
degree of shape coherence we can call it
say it again
no they don't they don't stay in close
proximity no no no the statement is the
volume of the region in phase space is
maintained so what that tells you is
that if two points do diverge or let's
say that if the phase volume stretches
in one direction it must compress in the
other direction okay so it's not
completely without rules but the only
real rule is that the phase volume is
maintained as time evolves volume volume
means the same thing that it does in
calculus it means the integral it means
break up the region into a lot of
extremely small cells let's let's talk
about area for the moment let's not try
to get the higher dimensions for a
minute let's just take the case of 1q in
1p then what I mean by volume of phase
space is just the area of the region
regions have area a regardless of
whether they're squares they don't have
to be squares or rectangles to have area
you have to use a little bit of calculus
to add all of these up but there's a
well-defined notion of area and as you
follow it the shape may change but the
area doesn't now if it's a higher
dimensional system with more than one P
and Q incidentally as always there's one
Q for each P they come in they come in
pairs but we simply have to think about
the higher dimensional version of volume
where we break up the volume into lots
of little cubes count the cubes and add
them up yes Michael
[Music]
yes absolutely
that's right nearby means close by in
position and momentum but it doesn't
mean that they stay close by okay and in
particular the chaotic case is the
situation where there's a great deal of
tendency for points which are close
together to each other to diverge from
each other but always at the expense of
compression in some other direction
yeah yeah all right
we're having which happen now oh you say
well yeah that's that's basically
friction you say you say there are many
ways to start then you can wind up with
a six on the table yeah
right if you didn't air friction we just
bounce off the table and and rebound and
it wouldn't stop yeah yeah right
friction is incredibly common right yes
yes and when you're doing a mechanical
system in practice if you're an engineer
you don't want to forget friction if
you're a physicist you often want to
forget friction you want to pretend it
didn't exist in fact in many situations
microscopic physics doesn't have
friction if you really follow every
degree of freedom of a system then
there's no concept of friction in the
equations friction is a collective
effect of many variables some of which
you just decide I'm going to ignore so
it's a thermodynamic idea rather than a
an idea of basic mechanics of course
they have to be consistent with each
other when you decide not to look at a
collection of degrees of freedom the
behavior of it may nevertheless reflect
the fact that in some bigger sense the
phase space volume is conserved our
energy conservation is one which you
might not realize if you are sliding
around things on the table so energy is
lost kinetic energy there's no potential
energy it doesn't go up or down kinetic
kinetic energy is just lost looks like
energy conservation is wrong those who
would believe in energy conservation
would make a prediction and the
prediction would be that if you very
minutely and carefully measure the
temperature of the table the temperature
of the table will go up a little bit and
if you know the specific heat of the
table then you can find out how much
energy was absorbed by the table etc etc
and and the conservation of energy will
tell you or will tell you that there
must have been a degrees of freedom that
you ignored the same is true of this
conservation of information if you
discover a system which looks like the
phase space is contracting it means
you've left things out of the system now
that's a that's an empirical fact in the
sense that it has always proven to be
the case but by now it is a fundamental
fact derived from quantum mechanics but
let's take it as a well we're not going
to take levels theorem as a given we're
going to prove it but we're going to
prove it from something we're going to
prove it from Hamilton's equations all
right so the beauty of the Hamiltonian
form of mechanics is this flow picture a
flow in phase space a very predictable
flow in phase space where the points
move in a very characteristic kind of
way and that characteristic way is
Hamilton
questions the entire flow is determined
by a single function of all of the P's
and Q's if you know that function then
you know the flow on the phase space and
if you started a little boat sailing on
the flow which means you start a system
at a particular point in the phase space
that Hamiltonian flow will tell you
where it is at any future time okay and
that flow is incompressible but let's
let's some first define the flow so this
is dynamics as a flow in phase space as
a kind of fluid flow
alright so every point has associated
with it a motion that depends only on
where you are and emulsion means a time
derivative this is P this is Q it means
a P dot and a Q dot in other words a
velocity but not a usual velocity a
velocity in phase space consisting of
the ordinary velocities which are the Q
dots plus the P dots which are the time
derivatives of the PS okay so we just
write down Hamilton's equations p sub i
dot is equal to minus D H by D Q sub I
notice they come in pairs is plus the H
by DP DP P sub I that's it that's all of
mechanics in a nutshell in the
Hamiltonian form okay all right let's
consider what it means for a flow to be
incompressible what we're all t'me going
to prove is that this flow is
incompressible what exactly does it mean
let's start with a one-dimensional flow
and see what it says so one-dimensional
flow just means you have a line and
let's populate that line with a uniform
distribution of points for simplicity
I'm taking a uniform distribution
means that the density of points is the
same everywhere along the line or
equivalently the little tiny separation
between points is uniform along the line
what kind of motion constitutes
incompressible flow well basically
there's only one kind and that's just
all the points move together with
exactly the same velocity what would
happen if the velocity varied from point
to point along here for example suppose
I went faster over here than over here
then there would be a clumping up of
points in between I think there will be
a clumping up and they increase in their
density all right but let's say let's
consider the mathematics of it what's
that yes yes but let's just talk about
flows for a minute I mean let's forget
phase space and just talk about the
general concept of an incompressible
flow and we'll come back to phase space
as a special case that's right you're
not you're right
but nevertheless it's possible to think
of incompressible flows in one dimension
the very trivial all of the you can have
any velocity you like but the velocity
has to be completely uniform all across
the line otherwise there's a clumping
now you can think of it in two ways you
can either say take a set of points and
follow them let's take a connected set
of points like this and follow them one
way of saying incompressibility is just
that the volume of that particular set
of points stays the same all right
that's a view of it following the points
there's another view of it which stays
at a particular place in space let's say
we tell you stay at a particular space
in place in space in other words a
little volume we're not going to move
that volume but points are going to come
into it and points are going to depart
from it if the fluid is moving to the
right of course it could be moving to
the right in some places in the left and
other places but let's count all
velocities as positive to the right if
if the fluid is moving then some points
leave this region and some points enter
the region to say it's incompressible
says that the number of points that
enters is the same as the number of
points that leaves in any given time
interval
what is the condition for that well it's
more or less obvious that if the
velocity let's imagine that the velocity
is fast over here and slow over here all
right then it's quite clear that the
number of points that are exiting the
region is proportional to the velocity
at this end right let's call this point
two and point one the number of points
exiting will be proportional to the
velocity of the points to the right and
that we can just call v2 that's the
number of points exiting the number of
points entering is the velocity at one
so the net increase shall we take
increase or decrease let's talk about
the net decrease the net decrease in
points in this volume here is going to
be proportional to V 2 minus V 1
actually it's V 2 minus V 1 times the
density of the points but as long as the
density is uniform then then we can just
write that the yeah that's the point
that's the point that's exactly the
point of the rate if the velocity yeah
if the velocity is higher at this end or
whichever way if the velocity is bigger
at one end than the other then the
density inside has to be changing
incompressible means that the density is
not allowed to change okay so what it
says is that V 2 minus V 1 must be equal
to zero that it cannot be a gradient or
a variation of the velocity this must be
equal to zero now if I take a small
interval let's let's imagine that this
is a small interval then the difference
of the velocity at two and one
assuming everything is nice and smooth
and differentiable that calculus applies
for a small interval we can say that the
difference of the velocities is just
proportional to the derivative of the
velocity with respect to x times some
Delta X where Delta X is the is the
interval here and then we would write
that this has to be equal to zero Delta
X is just a small interval it's not zero
it's just a number so it follows then
that the derivative of the velocity with
respect to X must be equal to zero for
an incompressible fluid which is what we
would intuitively imagine that the whole
fluid has to move together since all of
the motion is along the x axis I'm going
to write this as the derivative of the X
component of the velocity there is only
an X component of the velocity but just
to use a notation which will generalize
I'll write that this is DV X by DX is
equal to zero that's the condition in
one dimension for an incompressible
fluid and it is very trivial questions
good all right now let's move on to two
dimensions I'm going to call the two
directions X and y y naught x and y now
let's ask in a small interval of time
what is the increase or the decrease I
guess v2 minus v1 would be the decrease
in the number of points what is the
decrease in the number of points in this
little square here let's take the square
to be of size Delta X and Delta Y it's
not a square it's a rectangle how many
points are entering and leaving from on
the various sides of this rectangle all
right let's start with a number of
points coming in here again we're
imagining that the density is completely
uniform so the number of points per unit
area to start with to start with is
completely uniform if it's
incompressible then
the number of points per unit area must
stay the same must be exactly the same
after afterwards so okay so how many
points come in from the side here well
the number of points coming in this way
per unit time is clearly going to be
proportional to the X component of
velocity the y component of velocity a
velocity in the vertical direction will
not contribute the points crossing this
boundary here so the X component of
motion here VX at the left-hand end here
will constitute the incoming points
coming in from the left vertical side of
the square what about the number going
out oh there's another factor there's
another factor anybody know what the
other factor is Delta Y yeah the size of
the Y interval here obviously the bigger
the Y interval for a given velocity the
more points will enter there so there's
another factor which is just Delta Y now
how about the number of points going out
on this side that's also proportional to
V X but VX at the displaced point all
right so we have to move ahead a little
bit and subtract the points coming out
on this side exactly what we did over
here the answer is going to be the
derivative of VX with respect to x times
Delta Y times Delta X where does this
Delta X come from the difference of the
velocity here and here is the derivative
of V with respect to x times Delta X
alright the difference between the X
component of velocity on the right-hand
side of the square and on the left-hand
side of the square is there a derivative
of V with respect to X times Delta X we
have to multiply that by Delta Y to find
the net number of points coming through
this coming in - on this side - out on
this side and that's this
okay so this is the knit number of
points coming into the rectangle through
the vertical edges now what about the
horizontal Ekta edges the horizontal
edges things can also move in
but now the X component the velocity is
not relevant the particles coming in in
this direction only the Y component of
the motion is relevant on the horizontal
boundaries so how many particles come in
from the lower-end
well that's going to be V sub y at the
lower end times Delta X but then we have
to subtract off V sub y coming out of
the top end here in order to find the
difference of the incoming points and
the outgoing points on the bottom and
top so that will give us the derivative
of V Y with respect to what with respect
to Y because we're going from bottom to
top times Delta Y
the whole NIT change in the number of
points in this rectangle is simply
proportional to Delta X Delta Y that's
just the area of the little region
times a quantity which is called the
divergence of the velocity field below
this is the velocity field it can vary
from place to place but it's definitely
it's fixed this is derivative of VX with
respect to X plus the derivative of V Y
with respect to Y if this is positive it
corresponds to a decrease in the density
what if it's an incompressible fluid if
it's an incompressible fluid then the
net number of points coming in and going
out must add up to zero all right we can
forget the Delta x times Delta Y that's
just a number for a little area and so
the condition of incompressibility is
that the divergence of the velocity
field is equal to zero let's drop this
factor here it doesn't do anything for
us DV X by DX + DV y by dy is equal to
zero this is not a general rule about
all possible flows this is a rule about
incompressible flows that start with
uniform density and maintain the
uniformity of the density notice that it
does not say that the velocity can't
vary from place to place but it does say
that if the velocity is increased if the
X component of velocity is increasing in
one direction then the Y component must
be decreasing in the other direction so
it does not say that the velocity can't
change from place to place it doesn't
require the motion to be just rigid
motion are more complicated kinds of
things can satisfy this kind of flow but
it is constrained in this way ok
questions yes
yeah but we said we take an
instantaneous photograph of it then we
ask at this instant what's the time rate
of change of the number of points in
here so at any instant some something
which is coming in here yes it could it
could turn around and go up here that's
perfectly acceptable but that will be at
a later time that it'll be finally found
going out here so we tell you imagine a
snapshot instantaneously right now what
is the time rate of change of the number
of points in this volume okay so if the
point is coming through here it's not
going out there it replies to any number
of dimensions so in any number of
dimensions we could write there aren't
enough letters in the alphabet but I
some x y&z here just refer to a general
space having nothing to do with
mechanics just a general flow in any
number of dimensions the general rule
for an incompressible flow would be that
the derivative of the I component of the
velocity with respect to the i'f X is
equal to zero
all right that's also written in the
notation so some of I thank you that's
also written in the notation that the
divergence of V is equal to zero
divergence simply means this act of
differentiating the eigth component with
respect to the eigth coordinate and
summing them up it's called the
divergence yeah I assumed uniformity of
the density yes yeah if I hadn't assumed
uniformity of the density I just did
that to make it simpler and I now
assumed uniformity of the density what
you will have to do is invent the new
quantity which is the density at any
point times the velocity and then that
would be the thing which would have zero
divergence
but if the density is uniform then that
factors out so just for simplicity I
imagined that the and after all if it's
an incompressible fluid it stands the
reason that its density is everywhere is
the same since you can't change its
density
so let's come back the phase space what
are the X's the X's are just the
coordinates in some space in the case of
phase space the coordinates are the P's
and Q's the P's and Q's are the
coordinates so we should identify the X
is not just with Q but with a P and the
Q so the number of X's is twice the
number of Q's 1 1 X for each P and Q
that's the space we're dealing with if
there's only one Q then it's a
two-dimensional space 1 P and 1 Q and so
forth all right what are these
quantities over here P sub I dot and Q
sub I got those are the local velocities
in the phase space they are the V sub
i's I now goes from 1 to 2 n it runs
over all of the coordinates in the
momenta but these are we could call this
here we could call it V P sub I the
velocity along the axis P sub I this we
could call V Q sub I so P dots and the Q
dots are the things that are being
called the velocity now let's calculate
the divergence of the flow we're going
to calculate the divergence of the flow
and of course what we're going to use is
Hamilton's equations to see that the
divergence of the flow is exactly 0 all
right so the divergence of the flow we
take V P sub I and we differentiate it
with respect to P so
I so let's let's write the divergence of
the flow over here first of all we have
derivative of the P I'm not going to
bother writing the I Wow maybe I should
well let's let's let's just do it for
two dimensions first then we'll come
back and talk about it for any number
derivative of VP with respect to P with
only a one P and one Q what is that well
that is equal to the derivative with
respect to P times VP but VP is minus
the derivative of H with respect to Q
all right that's the first term what's
the second term in the divergence the
second term is just DV q by DQ alright
what does V Q that's over here plus the
H by DP so this becomes D by DQ of plus
D H by D P if we have several
coordinates and momenta then we would
put an I here in sum over I everywhere
also in the lower equation here I won't
bother with it though well now you can
see what happens
partial derivatives can be interchanged
the derivative with respect to P of the
derivative with respect to Q is the same
as the derivative of with respect to Q
of the derivative with respect to P the
P DQ is the same as DQ DP it doesn't
matter which order you differentiate
something in alright this one has a
minus sign this one has a plus sign so
these cancel in pairs they cancel in
pairs
in fact it's not just that the
divergence is equal to zero its equal to
zero in a special way where they cancel
in pairs but that's okay the divergence
to some of these DP that's exactly equal
to zero okay so the sum of these add
them together
that's exactly equal to zero and that
proves that the flow in phase space is
incompressible that is a quite a
profound theorem about about mechanics
the incompressibility of the flow the
fact that the phase points can't get
squeezed into a smaller volume that they
maintain sort of they have elbows and
they keep from running into each other
that is probably the most important fact
about Hamiltonian mechanics that is this
incompressible kind of flow let's
concentrate for a little while on two
dimensions one P and one Q remember this
does not say that the distance between
points stays the same as I said what it
says is that the volume of a region of
phase space stays the same so a region
like this could flow into a region that
looks like this
as long as the volume stays the same and
in particular two points which are
fairly close can get fairly far so
there's no conservation of the distance
between points in the phase space only
the volume only the volume of area of a
patch of phase space um this idea that
let's take the two-dimensional case that
area is the important thing and not
distance
you can see in another way let's take
the world's simplest mechanical system a
particle moving along one axis with an
energy which is X dot squared a kinetic
energy which is X dot squared over two
I have even gone so far as to set the
mass equal to 1 to make it simple all
right there's a canonical momentum piece
of X which is equal to the derivative
this is the Lagrangian Lagrangian is X
dot squared over 2 the momentum
canonically conjugate to X is the
derivative of the Lagrangian with
respect to X dot and that's just X dot
now I could have used other core another
coordinate I didn't have to use X to
describe the system what would have
happened had I use two X suppose I
defined a new variable let me call it Y
it does not stand for another axis it
just stands for another representation
of exactly the same physics
ah no coordinate I'll define and let's
call it alpha times X alpha is just any
number fixed could be 2 it could be a
half it could be 7 whatever it is this
is just another representation of the
same system where it's been labeled by a
different axis or an axis with either
closer or further points points are
different units it really comes down to
a unit transformation okay uh what is X
dot X first of all is equal to Y over
alpha and X dot is just equal to Y dot
over alpha let's rewrite the Lagrangian
the Lagrangian doesn't change or grunge
Ian is the same numerical quantity no
matter what coordinates we use the
Lagrangian will now become
Act is still X dot squared over two but
that becomes now y dot squared over
twice alpha squared I've just written
that X dot squared is y dot squared over
alpha squared that's the new Lagrangian
it's just another description of exactly
the same system what is the canonical
momentum conjugate to Y well its DL by D
Y dot piece of Y is equal to Y dot over
alpha squared okay let me now write
right over here we have y equals alpha
times X we have P y equals y dot over
alpha squared but that's also equal to X
dot over alpha I've used that Y dot is
equal to X dot let's see what if I
written yes Y dot is alpha times X dot
so wherever you see Y dot here just
stick alpha times X dot that cancels one
factor of alpha alright but this is also
equal to P sub X over alpha all right so
let me combine that with this equation
over here P sub y is equal to P sub X
divided by alpha notice what's happened
when you make it coordinate
transformation that stretches the x-axis
it shrinks the P axis sorry PR if you
make a transformation which stretches
the x axis remember now P of X and y are
not different directions of space
they're just different coordinates
describing the same thing this is a
coordinate transformation y is equal to
alpha X along with it
goes the transformation of the canonical
momentum but in the opposite direction
if you stretch the x-axis you shrink the
P axis okay that means for example if
you take the phase space let's we can
represent that either in terms of X or Y
and we take a little region let's call
this X and px suppose we represent that
same system in terms of Y and P sub y
well the the width of this box as
measured in Y units is alpha times as
big as it was in X units but on the
other hand the height of the region is 1
over alpha times what it was in X units
so whether the the height of this box
and the width of this box as measured in
either X or Y are not the same but the
area of the box as measured in x and y
are the same this is a characteristic
feature of classical mechanics that the
various coordinate transformations that
you would do if combined you do some
coordinate transformation but you also
calculate what happens to the canonical
momentum your what you'll find is that
areas in the phase space always stay the
same so the idea of an element of area
and phase space is very fundamental it
of course comes up in quantum mechanics
and quantum mechanics it's basically
roughly speaking the statement that
there's a smallest area in phase space a
quantum of area in phase space we'll
come to that but now I'll just mention
it now that things cannot be localized
in the phase space better than that
quantum of area how big is the quantum
of area funks constant no no not squared
not squared plus constant is not a
measure of Delta X or Delta P it's a
measure of area in the phase space
okay and it says that there's a
fundamental unit of area in the phase
space and that you cannot think of
anything smaller that's the uncertainty
principle yeah question absolutely
absolutely the mathematical statement is
that the time evolution of a system is a
canonical transformation yeah
that ditto did not what say it again why
does the why does it not conserve energy
no it looses can lose his potential
energy gains kinetic energy right good
okay so let's our the statement the
statement that the flow in phase space
flow in phase space is incompressible
that's called levels theorem where where
where here now this wasn't representing
the time dependence of anything this was
just two different sets of coordinates
that's all no well this is a canonical
transformation yeah Wow right
and we haven't we haven't discussed that
at the moment but but the answer is yes
that is a canonical transformation all
the interesting transformations of
mechanics are called canonical but we
haven't defined them yet so let's
[Music]
alright let's talk a little bit since
it's been raised about what chaos means
that's right means the area in quantum
mechanics it's of course connected with
the uncertainty principle that's right a
P dot and a Q dot a P dot and a Q dot
you want to multiply P times or Q dot ok
let's multiply a picnic you dot which
doesn't change is Delta P times Delta Q
the area of a little region of phase 2
out of phase space it doesn't change you
have you have X and P piece of X and you
have some other coordinates which you
can call Y and piece of Y and if you
calculate the take a little region of
area here break it up and calculate its
area by adding up the squares and giving
them an assigning them an area Delta P
that sorry Delta x times Delta P sub X
or add them up by assigning them Delta Y
times Delta P sub y you'll get the same
answer so it's the area element the size
of an area which is will be the same in
whatever coordinates you use whatever
coordinates and momenta if the
coordinates and momenta come from the
same Lagrangian ok so yeah so area area
has an invariant meaning in the phase
space
it takes on its sharpest physical
meaning when we get the quantum
mechanics where it really corresponds to
the uncertainty principle
well I what does it mean the fact that
it's conserved means that it's useful
that that
yeah-yeah-yeah-yeah-yeah it tells you
what the right probability measure is on
the phase space that invariant
probability measure on the phase space
and that's useful in statistical
mechanics yeah yeah and it for example
for example if you have some chaotic
system as a phase space so that means
the phase point moves around and gets
you know really moving around in some
wild way ah and you ask how often or
what is the probability that is found in
a little region of phase space then the
answer will depend only on the area of
the region of phase space so area would
be the thing which would tell you the
relative proper if you want were
interested in the relative probability
probability could mean the amount of
time spent in this little region here
the phase point moves around a little
goes crazy we stand there and watch it
and ask how much time does it spend in
that little volume that's a measure of
the probability that you would find it
in that volume versus this volume all
right the answer is that the relative
probability of being in two different
volumes of phase space is simply the
ratio of the areas for a chaotic system
yeah for a chaotic system yes right well
here I'm defining probabilities not by
expectation zero you're naive
expectations I'm defining it by the
amount of time integrated time that it
spends in a region so that's a that's a
stronger yeah right the the Yaak the the
amount of time that it's in here divided
by the total time that you sample the
system tends to a finite limit for
chaotic systems yeah now I know it
probably does but I have to think about
it no I'm not sure what the word
stochastic means frankly I think it just
means an element of randomness
what does stochastic was the word
stochastic mean none non deterministic
yeah well of course chaotic systems are
secretly deterministic but you can't
follow them very long in time whether
unless you have a degree of precision
which is far beyond what the what's
reasonable I don't want to I don't want
to get into chaos too much right now I
will just tell you very quickly one
other thing about chaotic systems or the
difference between chaotic and not
chaotic both of them have the feature
that the phase space volume stays the
same but that seems a little odd because
you would think that the systems which
are chaotic
you know just systems banging into each
other and random collisions not random
really but governed by the equations of
motion but nevertheless complicated
enough that they seemed to scatter all
over the place you would think that the
phase space volume increases increases
because you get because after a while
you know less and less about where the
system is well that's not exactly true
what here's the way it works
posing you take all right so the correct
statement is that this little sphere in
phase space will evolve and it will
evolve into something which may look
more or less like a sphere but it may
also look you know grow all kinds of
complicated fibers and become very
fractal ated I don't know that it is
fractal ated a word probably not you
know I start to grow okay now here's
what you do
you I want to put some more in
okay etc all right now if I literally
mathematically follow the points
incidentally this will be just as
continuous as this if you look on it
under a microscope you will discover
that this phase space is completely
continuous nothing discontinuous is
broken off it or anything like that but
as time goes on it tends to get more and
more fractured like this let's just call
it fractured if you take each point each
point goes to a definite point if you
follow the volume you'll find the volume
is the same but now you can do something
which is called coarse graining coarse
graining means a purposeful decision to
draw little circles or spheres around
face points and roughly speaking what it
corresponds to is saying I don't have
enough precision to be able to
distinguish points of phase space
strictly speaking I can only distinguish
this little volume from that little
volume from that little volume in that
little volume so we do is draw little
spheres around each phase point and
follow the little spheres rather than
the individual phase points follow the
spheres and what will happen is this the
sphere may go over here the sphere may
go way over here this one may go way
over here this one may go way over here
but if we were to take the total area
occupied by these spheres I think it's
pretty obvious that the total area will
be much larger than the area that you
started with
if this thing branches out and fractal
eise's it is so the coarse grain
following of the phase space allows the
area to get bigger and bigger and it
does get bigger yeah
[Music]
yep yeah because of the coarse graining
is effectively random in other words if
you take the coarse-grained phase-space
the nasty thing that happens is no
matter how small the coarse graining is
if you wait long enough points within
one coarse grain here will separate and
get far outside acquire coarse grain if
the coarse grain is defined by little
spheres so what's that
nono though they're not the mapping from
the first no the mapping from the no
Lauro they're not the mapping from the
first they're just spheres in the phase
space again spheres that you can
distinguish by the coarse-grained
apparatus that you have in the hand so
the coarse-grained apparatus that you
happen to have defines a coarse graining
on here and it's not the same as the
coarse graining that would inherit by
following this coarse graining it's just
what you can measure what you can
distinguish and of course what you can
distinguish depends on the precision of
your of your measuring apparatuses and
so forth but no matter how precise your
apparatus is in your description is no
matter how small a coarse graining here
eventually points within a single cell
here of this coarse graining will
diverge and get outside the the
corresponding coarse graining
description of what happens later that's
what happens to a chaotic system most
systems are chaotic of course the the
non chaotic systems which sort of
maintain
their integrity without in the
coarse-grained sense those are rare and
exceptional yeah say it again
yeah change in time yeah the area of the
of the core schedule you take the
original region of phase space you
figure out exactly what it does and then
in the original region of phase space
you draw spheres if the original region
is a nice simple sphere or something
like a sphere then when you add up all
of the of these coarse-grain spheres
you'll get the volume of the original
sphere back pretty much but now you take
this tree this very very finely fractal
iced tree that it grows into which may
fill up this big piece of this
blackboard with very fine filaments very
very close to each other okay so for
example if there are filaments extremely
close to each other so close to each
other that the course greening so the
slops over or slots over the distinction
between these different filaments then
the total volume of these coarse-grained
green spheres will in general exceed the
volume that you started with why yeah
it's basically just going to cover this
hole the whole tree there yeah yeah
you're just yeah yeah yeah so you just
look at this picture with fuzzy eyes and
you see how big how big this region is
this region can be much much bigger yeah
yeah Wow lots of ways any what yeah
you're counting everything within this
little sphere here even though the fiber
in here may be much thinner than the
sphere but the fibers are so thin that
your apparatuses cannot tell exactly
where you are in the sphere all you know
is you're somewhere in the sphere so at
the end of the day if you ask where am i
if I start in here the answer to the
question is you're somewhere is in here
in a much bigger region if you have
infinite precise control over the either
the mathematics or the measuring
measurement then the volume of this
fibered ultrafine filamentary structure
would be exactly the same as the
original ball here but you don't follow
things that way that's not the way you
follow things in an experiment you say
okay I take the I detect upon a particle
somewheres in this phase region over
here I can't tell exactly where it is
because it requires too much precision
so in that sense the phase space volume
does expand
it's called coarse graining and it is of
course the origin of the second law of
thermodynamics okay there's one other
point that I should emphasize I don't
know when when and if we'll come to it
in a more
away but the entropy of a system is a
measure of the volume in phase space
that the system occupies are the
detected detectable volume in phase
space meaning to say if you know about a
system that it's somewheres in here you
take that volume of phase space that
volume of phase space which corresponds
to the boundaries of the region that
that your knowledge tells you that the
system I'm getting them getting tied up
you have some knowledge about the system
knowledge of the system is that it's
somewhere is in here in this volume of
phase space the entropy is a direct
measure of that volume in fact it's the
logarithm of that volume essentially
that's that now the system evolves and
what you know afterwards is that it's
somewheres in here you're not either
clever enough or precise enough to be
able to distinguish these fibers so all
you know is that the system is somewhere
is in here the phase space volume is
bigger afterwards than it started in the
beginning because you haven't followed
the detail sufficiently carefully to
know precisely to distinguish this bit
of fiber from that bit of fiber that's
why entropy increases basically it's a
statement that when the phase space
fragments like this you lose information
for not because information is really
lost but because you have a hard time
following the system it's a
technological thing you can't follow
things with the sufficient precision
okay
that's a that's liouville's theorem
as I said it is extremely fundamental to
both classical mechanics and its
evolution into quantum mechanics what I
wanted to do next
completely different direction was to
work out the equations of a particle we
have not studied very many examples
incidentally you'll notice I want to
study the example of a particle moving
in an electromagnetic field there's one
thing that's new in a magnetic field
that we haven't encountered yet and it's
the existence of velocity-dependent
forces so far the forces that we've
imagined depend on where you are but not
how you're moving or we said that
there's a potential energy function U of
X and we differentiate it to find the
force well the left hand side is just a
function of position so the force on a
system just depends on where you are
there are forces in nature which depend
on velocity magnetic the magnetic field
acting on a particle is a prime example
incidentally friction is also an example
friction is an example where the
velocity of where the force on an object
does depend on its velocity if it's
standing still is no friction force if
it's moving there is a friction force
typically the faster it moves the larger
the force there's a fundamental
difference between magnetic forces and
friction forces and the simplest way to
say it is that forces due to magnetic
fields although the velocity dependent
are de Rivel from a principle of least
action they have a Lagrangian
formulation there's a conservation of
energy and there's a Hamiltonian
formulation it's just that the
Lagrangian and Hamiltonian have a new
little twist to them that
that will that I want to work out now
forces due to electric fields are no
different than the forces we've already
studied I think I'll come back to
electric fields I want to concentrate on
magnetic fields because there's a new
phenomenon there that that's interesting
all right so what is the force of a
particle if it's in a magnetic field
magnetic field is called B it's a vector
and it depends on position B of X the
magnetic field what is the force of a
charged particle on a charged particle
the cross B well actually Q times V
cross B where Q is the charge of the
particle velocity cross magnetic field
it's a vector the cross product of two
vectors both of these are vectors this
is the velocity of the particle this is
the magnetic field and so it says well
everybody know what a cross-product is
assume everybody mom yet that means yes
or no yeah right right let's see write
something to do with the thumb
I'm going to assume everybody knows what
a cross-product is although I will
define it okay so let's define the
cross-product right now the
cross-product can most easily be defined
through its components V has three
components x y&z B has three components
x y&z a cross-product has three
components here is the role the Z
component let's forget Q here let's just
write the rule the Z component of this
let's call the Z component of it V cross
B the Z component of it is equal the V
xB Y minus dy DX
I remember one one of them I remember
the Z component and then if I want any
of the other components I just cycle Z
goes to X goes to Y so the next one
would be X V cross B the X component and
I cycle X goes to Y dy Y goes to Z DZ -
minus the z dy and last one V cross B
and the Y Direction is equal to X Y Z Y
Z X minus V X B Z that I get that right
I think I got it right that's the
definition of the cross product
now we need one of the concept we need
the concept of the vector potential the
vector potential is simply a way of
writing magnetic fields but you would
think well if it's just a way of writing
the magnetic field why not just use the
magnetic field and the answer is you
cannot write the mechanics of a charged
particle just in terms of the magnetic
field you need the vector potential
or you can of course write the mechanics
F equals MA but you cannot write a
Hamiltonian you cannot write a
Lagrangian you cannot write even an
expression for the canonical momentum
you need an auxiliary quantity called
the vector potential and the vector
potential is just defined by the
condition that the magnetic field is the
curl of the vector potential let's write
down with curl means because we'll need
it later the vector potential is also a
vector the magnetic field is a is a
vector the vector potential is a vector
what's special about things which are we
are curls of other things they have no
divergence and the magnetic field is a
thing which has no divergence and that's
why it's convenient to write it this way
but for our purposes let's imagine that
the vector potential is something
fundamental that we begin with and that
the magnetic field is a derived quantity
we're going to see that the mechanics of
a charged particle is best written in
terms of the vector potential not the
magnetic field
okay let's write down the definition of
cross product the cross product again
can be defined by its components go
across a Z component of it now it's
basically the same thing except wherever
you see a V up there wherever you see a
B stick an A and wherever you see a V
over there put the an upside down Delta
alright so this just says that this is
what is what is Delta mean it just means
the collection of three derivatives
Delta X Delta Y and Delta Z adjust
derivative with respect to X the
derivative with respect to Y and the
derivative with respect to Z I assume
you all know that kind of thing a little
bit of vector calculus but let's just
write down exactly what this means this
means derivative with respect to X of a
sub Y minus the derivative with respect
to Y of a sub X and then cycle it down
del cross a the X component of it is the
Y derivative of a Z minus the Z
derivative of a Y and just cycle it down
X goes to Y Y goes to Z Z goes back to X
I won't write them all down so we can
now write that the force on a charged
particle due to the magnetic field is
the charge times V cross B which is also
the charge times V cross the curl of a
I'm going to bore you a little more with
this by writing out the individual
components tour by writing out the Z
component of force here there's a little
bit tedious but we'll need it
we'll need the expression for the Z
component of force explicitly in terms
of the vector potential so let's write
it out this is just an expression this
is just an exercise in substituting
alright
I want the Z component of the force F
sub Z is equal to the charge now the
cross product and we have a cross
product where is it we have the Z
component of V cross B that's V xB Y
minus V Y the X but now let's just
substitute in for B Y where is it oh
brother this is Q V X now go to
hmm B Y is d Z ax minus D X a Z that's
the first term minus V y times BX which
is dy AZ minus DZ a Y well this is a
fairly complicated expression for the Z
component of force but if you know the
vector potential you can work out what
this is it's just some vector field you
can multiply it by the velocities and
you can compute the force the Z
component the force if you want the
other components again you just cycle
the X X goes to Y y goes to Z you just
cycle through alright that's the force
on a particle and the equation is just
Newton's equation F equals MA but now
the new thing is that the acceleration
doesn't only depend on the position it
does depend on the position through the
through the position dependence of a but
it also depends on the velocity so these
are called velocity dependent forces
because they depend on velocity we want
to formulate the theory of a charged
particle moving in a magnetic field in
the Lagrangian or Hamiltonian form not
obvious that we can do it it is not
obvious that this is of the form that
that allows us to have a Hamiltonian
Lagrangian and so forth or an action
principle the easiest way to confirm
that is just to make a guess at what the
action is and then follow it through the
new thing in the action well first of
all we do expect that if the electric
charge was zero that there would be the
usual terms
alright so let's write the action let's
make a guess for the action the integral
of the Lagrangian DT well first of all
guess has the usual MV squared over 2 D
T where V squared means the sums of the
squares of the X component Y component
and Z component of the velocities the
usual thing this is what we would do for
a free particle and if the particle has
no electric charge it doesn't feel the
magnetic field and then there's another
term and the other term is as simple as
it could possibly be we take the orbit
of the charged particle we break it up
into little segments each segment we
call the X also dy and DZ but let's just
abbreviate it by calling it DX each
little segment defines a differential
displacement in the X dy and DZ the
added term that's first of all
proportional to the electric charge if
there's no magnetic field then obviously
there is no additional term even if
there is an electric charge the extra
term is related to both the charge and
the magnetic field but it's related
through the vector potential wind with
vector potential did the probe or do I
erase the most important equation I did
okay I assume you all memorized what V
cross B looks like in terms of the
vector potential and you'll recognize it
when I write it down
I didn't mean to erase it all right what
can you write down it's going to have
the vector potential in it so let's call
it a sub I the simplest thing you can
write down is the integral the line
integral of the vector potential dotted
into the differential displacement DX so
in other words what this means is every
place in space you have a vector
potential as the particle sweeps out
through its trajectory there's a
contribution to its action from each
little differential displacement here
and the contribution to the action is
just take the dot product of a vector
potential with the displacement in other
words the component of a vector
potential along the direction of the
displacement and add them all up
very simple prescription how do I know
that's the answer well I know that's the
answer because it was written down in
the early parts of the 20th century and
I learned it as a student yeah
the X I good and or you can just write a
dot DX where you can think of both a and
DX as little vectors or one is a little
vector the other is just a vector
alright a dot DX can represent the dot
product of the vector potential with the
differential displacement how do you
know what the units are you know the
units of a are what the units of Q are a
time but how can you tell me what the
units are unless we know the units of Q
R or a I don't know what the units of a
are well units of a in the units of Q
acted obviously the combination Q times
a is the important thing and the
combination of Q times a has units that
are required to make this have the same
units as that
what are the units of an action
incidentally or momentum times position
sorry momentum yeah momentum times X no
problem you can always give Q the right
dimensions this is correct in some
appropriate choice of units this is
correct and it depends on the unknown on
the conventions and some conventions
there are pies floating around or speeds
of light but you can set them all equal
to one and since I'm trying to describe
a phenomenon rather than a numerical
effect let's just set all in numerical
constants equal to one or two absorb
them into the electric charge yeah
okay now that this is yeah yes that's
exactly what I was about to do this does
not have the usual form of an action
which depends on velocities and
positions the A's depend on position as
a rule but what does this DX here all we
have to do is divide and multiply by DT
DX by DT that's just the X component of
the velocity so we can also write this
as our a dotted into the velocity
integral DT or I can write it in terms
of components a sub I X sub I dot DT and
now we see the usual form the what's
that some of I yeah some of I all right
so we see and this of course is also X
sub I dot squared sum over I not going
to write it all right so we have then a
typical Lagrangian we can write out now
what the Lagrangian is and it depends on
position and velocity the way any
respectable Lagrangian word Lagrangian
for particle in a magnetic field
is 1/2 the mass of the particle times X
sub I dot squared where I goes over XY
and Z some plus Q the charge on the
particle X sub I dot times the vector
potential a sub I what's the next step
the next step is to prove that the
equation of motion for this object is F
equals MA with the right hand side being
where is it QV cross B F equals Q V
cross B if we can do that if we can now
take the Lagrange equations here's the
Lagrangian take the Lagrange equations
and see if we can prove that the force
on a particle this is equal to mass
times acceleration that this is equal to
QV cross B if we can do that then we're
finished we found the Lagrangian
formulation of particle and a magnetic
field so this is now an exercise and
just plugging its row I could tell you
to go home and do it but let's do it
let's just carry it out there's one
curious thing coming up with that we'll
see in a second first of all what is the
canonical momentum piece of X well it's
the derivative of the Lagrangian with
respect to X dot so let's take the X
component I now mean the X component we
could write down y&z also that's going
to be M X dot from here usual thing
normal momentum but then there's another
term plus Q
a sub X and of course the same kind of
thing for py and PZ so we have a system
now where the canonical momentum is not
just M times V it's n times V plus
something proportional to the vector
potential
sometimes M times X dot is called the
mechanical momentum it's sometimes
called the mechanical momentum meaning
to say you know its meaning to say
nothing other than that its mass times
velocity and the whole thing is called
the canonical momentum all right so
that's that's one new thing here we see
something new that the there's a
discrepancy between the canonical
momentum and what we naively think of as
momentum but now let's work out the
equation of motion let's work out the Z
component we can all three of these but
let's get the PZ is equal to em Z dot
plus QA sub Z I'm going to work out the
Z component the core of the equation of
motion what is it it's the time
derivative of the momentum or just right
now over here D by DT of the L by D Z
dot is equal to DL by DZ the L by D Z
dot that's just PZ ok so we have to take
the time derivative of this thing here
let's write down the time derivative its
M Z double dot that's mass times
acceleration and then there's another
term in the time derivative of PZ it's
plus Q times the kind derivative of AZ
now in a moment we'll come back and work
out with the time derivative of AZ is
but let's go to the right-hand side of
the equation the right-hand side of the
equation
is the derivative of the Lagrangian with
respect to Z what is it that the pens on
Z in here X dot those are Z dot doesn't
depend on Z the a a and all of the A's a
X a Y and a Z all depend on on on Z in
general so let's go through them Q now
the first term here would be X dot times
a sub X so we would get a term which
would be X dot times the derivative of a
sub X with respect to Z remember we're
taking derivative of the Lagrangian with
respect to Z the only thing that depends
on Z are the three components of a let's
start with the X component X dot times a
X and differentiate it with respect to Z
the next one is plus y dot times the
derivative of a Y with respect to Z and
what's the last one Z dot times the
derivative of a sub Z with respect to Z
okay all I did was differentiate all
three terms here with respect to Z each
one of them depends on Z through the
components a what about the left hand
side what do I do with a sub Z dot let's
suppose incidentally that it's just a
static magnetic field magnetic fields
not changing with time it's just a fixed
magnetic field does that mean that this
is zero if it's a fixed magnetic field
no no because the particle moves through
the magnetic field I don't mean when I
say it's a fixed magnetic field I don't
mean it's the same everywhere in space I
mean it doesn't depend on time magnetic
field varies from place to place
particle moves through it and because
the particle moves through it there's a
time dependence of the magnetic field of
the vector vector potential at the point
of a particle so how do we calculate
that well we simply write em Z double
dot plus Q now AZ depends on X the AZ by
the x times X dot plus da Y sorry what
are we doing here yes sorry the AZ so
Xing AZ we're calculating AZ dot which
is derivative of a Z with respect to x
times the time rate of change of X plus
the AZ by D Y times y dot plus d AZ by
DZ times Z dot
that's equal to what's in the bracket
over here equals this okay I'll let you
stare at it for a minute remember the
right hand side came from explicitly
differentiating the Lagrangian with
respect to Z the left hand side came
from differentiating with respect to T
but notice they have similar form each
one here has a velocity times the
derivative of a on both sides so maybe
they combine together in some nice way
well first of all you'll see that this
term and this term are the same z dot
times da Z by DZ they're the same so
they cancel let's get rid of them now
what's left let's look at what
multiplies X dot X dot has the ax by DZ
and then when we shift to the other side
it will have what the AZ by DX so it's
going to have what sir let's move this
over now I forgot what it was plus y dot
what was it da y by DZ yeah possibly Y
with respect to Z right yeah okay now
what the term which multiplies X dot
when I transpose it it's going to have
minus d AZ by DX
oh that's starting to look familiar and
when I transpose the term that has Y dot
in it that will have minus d AZ by dy
I've transposed this term over to the
right hand side and I now have M Z
double dot or mass times the Z component
of acceleration is equal to this mess
over here well what is this mess this
mess is not so complicated it's got
linear terms in the velocities and what
are these objects the magnetic field the
curl of the vector potential in fact
this one is a X Y I think this is B Y is
a py a minus B Y and this one I think is
minus B X the X dots and Y dots are the
components of the velocity if you go
through it or just go back in your notes
to where I wrote it down in the
beginning this is just the Z component
of V cross B velocity hmm yeah okay
write the x times DZ ax minus DX AZ and
so forth yeah okay so you see that first
of all there does exist a Lagrangian
formulation here it is per particle in a
magnetic field you need to know the
vector potential first question is the
vector potential unique in terms of the
magnetic field no you can add things to
the vector potential we'll come to that
but not right now but if you know the
vector potential or you know a
particular expression for the vector
potential then you can write down the
Lagrangian you can write the Lagrangian
depends on the vector potential but the
equation of motion only depends on the
magnetic field so even if there is an
ambiguity in the vector
potential it doesn't affect the equation
of motion itself now we have an example
of a velocity dependent force a force
which is V cross B it's almost like a
friction force it is different the
friction force acts along the direction
of the velocity okay if an object is
moving in this direction the friction
force is along the same axis what about
V cross B because V cross B act along
the direction of the velocity no because
V cross B is perpendicular to both the
velocity and to B so then and this is a
big difference a big difference whether
the whether the velocity dependent force
acts along the direction of velocity or
perpendicular to it if it acts along the
direction of velocity it will obviously
have the effect of either slowing it
down or speeding it up depending on
which way it's acting all right but if
that's perpendicular to the velocity
it'll simply change the direction of the
velocity if the force is perpendicular
and it won't change the energy so energy
conservation will we can work out now we
can work out what is the expression for
the energy we do that by calculating the
Hamiltonian and when we've calculated
the Hamiltonian we then have from
previous theorems the theorem that says
that energy is conserved alright so we
can work that out let's work it out I
think we're I think we'll just that'll
just about take us to the end of the
hour oh excuse me there's something I
need to tell you I won't be here for the
next two lectures now I would like to
make up one of them I think the question
is what is a good time it could be this
week yet but it can't be next week
and it could be the week after but not
on Monday yeah we can do it this week
the septa does that worked out for
people if we would do it say Thursday
Friday is a bad day for me or we can put
it off we can put it off till I get back
and and just double up the week that I
come back I'll be gone for hmm
should we wait until Thursday how many
cannot think at Thursday well that's all
that's almost half the class or
supposing I were to put it off I'll be
back on the 11th now the 11th is
probably a Monday uh no 11th is a
Tuesday hmm yeah I was thinking about
Wednesday the 12th I think it is this is
12th or 13th does that work no I know
that
hmm well we can stay on Monday and there
but ya know we'll continue will continue
but at some point though we we run into
the next quarter I mean you know playing
you know points have to move rigidly you
can't all I could do that another thing
I have to do as I'm supposed to fill in
the missing lecture from the first of
the first lecture I was thinking of
doing that this week but the no no no
they we didn't miss a lecture but they
didn't they didn't film it okay what's
that
no that doesn't have to be no no I'm
happy that this is fun for me I just do
it for the fun of it no you know I
intend to keep going until the until
sometime near Christmas but we will miss
the next two lectures so I thought I
would try to make up one what's that
yeah two Monday's but then we'll be back
on Monday well we will be back on Monday
and we'll see if we need if we need to
fill in if we need to fill in with
another lecture we'll just figure out at
that point went to fill in well I don't
think we will do Christmas Eve let's
yeah yeah we go well look why don't we
just decide this when I get back on the
on the 17th we'll try we'll see when we
can fill in an extra lecture and I think
that will pretty much bring us to where
we want to be I mean I'm pretty
comfortable er three weeks yes three
weeks from today right right right right
next lecture is the 17th okay let's try
to just finish up what I was talking
about here I thought I think I finished
energy yeah okay let's do energy so all
we have to do is calculate the
Hamiltonian incidentally I'm assuming
that the vector potential has no
explicit time dependence in other words
equivalently that the magnetic field is
constant in time in that case the
Hamiltonian will be time independent and
there will be a conserved energy the
conserved energy is surprisingly simple
when expressed in terms of the velocity
is a little more complicated when
expressed in terms of the momenta so
let's work it out let's first calculate
[Music]
a piece of X as we saw before was n X
dot plus Q a sub X same thing for y and
z and I won't bother writing them down
separately okay the first thing we have
to do when we calculate the Hamiltonian
actually we don't need to we can just
what's let's calculate it not in terms
of P but in terms of X dot let's first
do it by calculating it in terms of
velocities to see what the expression
for energy is in terms of little
velocities all right so what do we have
to calculate we have to calculate P sub
X X dot plus P sub y y dot plus P sub Z
Z dot minus the Lagrangian right
that's the definition of age think each
momenta multiply it by the corresponding
velocity and then subtract the
Lagrangian that's a very general rule
that we derived all right if I plug in
for the peas their expression in terms
of velocities I'll have an expression
for the energy in terms of velocities
that won't be good for Hamilton's
equations but it will be good for
telling me what the energy is in terms
of velocities so let's do that let's do
that first P sub X times X dot that is M
X dot plus Q a sub X times X dot
likewise for y&z and I won't write them
out then we have to subtract off the
Lagrangian minus L so subtracting off
the Lagrangian means subtracting off
minus MV squared or MX dot squared I'm
concentrating on the X term and like and
X dot squared divided by 2 and then what
else minus Q X dot a sub X well that
gives me MX dot squared over 2 usual
thing no difference there
and look at this QA X X dot minus QA X X
dot they cancel completely so the
expression for the energy is exactly
just the original kinetic energy just MX
dot squared over two
likewise for y&z the magnetic field does
not contribute to the energy when
expressed in terms of velocities it's
just one-half MV squared in the story
now why is that why is that that's
equivalent to another statement it's
according to the statement that magnetic
fields do no work what's that yeah
that's right it's because the force is
perpendicular to the velocity so when a
particle moves in a magnetic field the
force is perpendicular to the velocity
the force being perpendicular to the
velocity means that the magnitude of the
velocity doesn't change alright a force
perpendicular to velocity deflects an
object but doesn't change its velocity
and so as the particle moves along its
direction may change but its speed
doesn't so to rid the original
expression for its energy in terms of MX
dot squared is just the full energy
magnetic fields do no work which is the
same statement as a statement that the
energy when expressed in terms of
velocities and so forth is exactly the
same as it was in the first place
one-half MV squared is conserved and you
know how particles move in a magnetic
field for example if there's a uniform
magnetic field into the blackboard
particles move in circular orbits with
uniform speed so the kinetic energy is
conserved
that's statement number one but in this
form you cannot use the Hamiltonian to
write down Hamilton's equations you have
to express it in terms of in terms of
the momenta but that's
easy all we have to do is solve for the
X dots in terms of the momenta that's
easy
X dot is just P sub X minus Q a sub X
divided by M so we can write down
immediately what the Hamiltonian is in
terms of peas instead of in terms of X
dots and it's just equal a piece of X
minus Q a sub x squared over m squared
over 2m squared excuse me I've just
substituted for the velocity their
expression in terms of the momenta yeah
note that it accelerates oh it does it
does it does but radiation is a higher
order effect that there's only important
when the particle is yeah it does it
certainly does radiate right but thus
far we have not they even tried to think
about radiation in order to understand
radiation we will have to formulate
Maxwell's equations and formulate them
in a Hamiltonian form and discover that
the radiation field itself has energy
yeah right so yeah this is the radiation
'less approximation
Mini's yeah sure absolutely
plus zyy turn right so it's a little bit
odd and a little bit surprising that
when expressed in terms of velocities
the energy just doesn't change but when
expressed in terms of canonical momentum
the expression for the energy does
change and now if we applied Hamilton's
equations to this expression here we
would again get precisely the same
[Music]
equations of motion what's that yeah
yeah all you do is replace the canonical
momentum omentum by the momentum minus Q
a sub X in other words this is just a
mechanical momentum here the NX dot term
the mechanical momentum and the
canonical momentum differ by Q times a
the energy is just MX dot squared so
it's just a mechanical you know just the
the square of the mechanical momentum
but you substitute the canonical
momentum if you want to do Hamilton's
equations so here's an exercise take
this Hamiltonian work out Hamilton's
equations and check that you get the
same equations of motion mass times
acceleration is equal to V cross B
that's a that's a homework assignment
the preceding program is copyrighted by
Stanford University please visit us at
stanford.edu