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let's proceed on with Poisson brackets
phase space has a structure to it it's
not just a bunch of P's and Q's but its
P's and Q's with a certain kind of
structure spaces have structures the
structures are the things which are
invariant under the various
transformations that you might want to
do an example of the structure that can
be on a space if you know anything about
curved spaces and differential geometry
and so forth the metric structure of
certain of metrical spaces of riemann
spaces are part of the structure of
those spaces they're characterized by a
metric on the space a distance formula
between neighboring points poisson
structures are different than metric
structures they're more abstract and
they were invented largely to describe
the structure of phase space and its
properties its properties on the
transformations which change the
coordinates now the coordinates of the
phase space are not just the Q's the
coordinates of the phase space of the
Q's and the peas and roughly speaking
the question the kind of question that
you might ask is what are the various
transformations that you can do for
example it mix up P's and Q's that
preserve the basic structure of
classical mechanics as I said this is
the kind of question the kind of
structural question a kind of formal
question that the French mathematicians
were awfully damn good at it turned out
to be very important to the future of
physics wasn't the French who figured
out why it was important there was a
French who figured out how how the basic
structure at the French and the and
Hamilton basically who figured out what
the structure of classical mechanics is
and it got more and more abstract the
most concrete version was of course
Newton's version
the most abstract version I don't know
how abstracting you probably get
infinitely more abstract than anything
I'm likely to teach you but the most
abstract end of it for our purposes is
the Poisson formulation of classical
mechanics based on the idea of plus norm
brackets what Poisson brackets do or
what they but they useful for is
describing flows on phase space one kind
of flow on phase space is the motion of
the phase space with time or the motion
of the points in phase space with time
how the actual points evolve with time
under the influence of a particular
Hamiltonian that's one kind of flow on
phase space but there are other kinds of
flows on phase space that you can
imagine with possibly somewhat different
purposes for example we can talk about
symmetries now we've talked about
symmetries the basic symmetry that the
problem for me and explaining symmetries
is not that that's hard to explain it's
that there aren't too many examples in
classical mechanics that allow you a
wide range of different situations to
describe the basic symmetries of most
classical mechanical systems our
rotations about various axes and
translations are special special systems
with particularly simple mathematics
sometimes have some hidden symmetries
that you might not have expected but the
basic symmetries are just those the
motions of space translation rotation as
you move on to more advanced physics you
discover more symmetries different kinds
of symmetries symmetries among
elementary particles and so forth but
those enough for us now what do those
symmetries have to do with flows flows
in phase space well let me begin with
flows in just coordinate space X space
we can think of a rotation we can think
of rotations in space as defining a kind
of flow in the space just imagine going
from here to here
taking our coordinates in space by
imagining an infinite number of
intermediate configurations and imagine
in your head going from one
configuration to another configuration
of the axes of the axes of a space
continuously or we don't have to imagine
motion of the axes we can actually
imagine motion of a system where we
rotate the whole system where you keep
the axes fixed and rotate the whole room
that rotation might not have anything to
do with the actual time dependence of
the system it simply allows us to
describe mathematically what a rotation
is a rotation is something which is
built up out of lots of little rotations
basically and so we can think of the
idea of a coordinate transformation
which rotates the coordinates of space
as a kind of flow likewise the
translation of space the motion from one
point moving our coordinate axes not
rotating them but translating them we
can also think of that as being built up
a given translation as being built up by
a lot of little translations which
define a flow
it's those flows in phase space not just
in coordinate space but transformations
which mix together or potentially could
mix together both the peas and the Q's
into a Richard structure then just the
transformations which mix up the
coordinates are used coordinates to mean
Q's or X's I use phase space to talk
about the peas and the Q's we can become
interested in we will become interested
in transformations which mix the P's and
Q's and so it's a richer family of
transformations that we can imagine and
those flows which move things around in
phase space in various ways are
described by the method of Poisson
brackets so I try to make it a habit of
not beginning in the middle even though
I may have covered something fairly
completely in the previous lecture so I
like to go back over
even if it means I do with two or three
times let's just review what we know
about Poisson brackets okay but let's
let's be French mathematicians let's
abstract rather than write down a
concrete die rather the concrete
definition of a Poisson bracket let's
think about it from a more abstract
collection of postulates or axioms about
plus norm brackets the Poisson brackets
between what I want are Poisson brackets
what go into the Poisson brackets a
thing I'll call a and the thing that
I'll call be both a and B are imagined
to be functions of the P's and Q's so we
can write plus own bracket of a of P and
Q with B of P and Q first of all has the
abstract property that it's
anti-symmetric anti-symmetric means that
it changes sign if you interchange a and
b that's the first abstract property
oops minus sign thank you most important
thing the minus sign right so that's
abstract property number one I've
described property number two I won't
write down in detail unless you ask me
to
it's the linearity all let me write it
down supposing we have the Poisson
bracket of some combination of a and B
let's say alpha a alpha and beta are
numbers any numbers alpha A plus beta B
plus some bracket with C is just equal
to alpha times the Poisson bracket of a
with C plus beta times the plus own
bracket of B with C
so that's the property of linearity this
is anti symmetry this is linearity then
there's a product rule capital a what
kind of what functions functions of P's
and Q's P squared plus Q squared sine of
P whatever not necessarily there's no
reason why in these formulas you can't
take alpha and beta to be complex
numbers you can have a oh yes you can
add A's and B's absolutely
you can add A's and B's just adding the
functions yeah you can certainly a days
and B's they're just functions are in
classical mechanics there isn't much
reason to be thinking about complex
valued functions but there's no reason
that you can't and sometimes you do
sometimes you do sometimes it's useful
so they're just general functions of the
coordinates of momenta remember the
coordinates when I write down P's and
Q's I don't just mean one p in one Q I
mean in PS and in Q's where n is the
number of coordinates of the system okay
so that's the linearity then there's the
product rule the Poisson bracket of a
times B with C is equal to a times the
Poisson bracket of B with C
Plus be
not a thank you B now I've written these
formulas in a particular order here I've
left the a on the left here and the B on
the right there's no reason for that
these are just numbers that is to say
they're functions whose values are just
numbers and it doesn't matter whether
you multiply a times the Poisson bracket
or the Poisson bracket times a
mathematically speaking these things all
commute which means you can multiply
them in any order I've written it in a
form which won't mean anything to you
now but it will mean next quarter when
we study quantum mechanics and quantum
mechanics things don't commute and I've
written in a particular order which is
still correcting quantum mechanics but
for our purposes now a times the Poisson
bracket of B and C is just nothing but
Poisson bracket of B and C times a so it
doesn't matter whether the B's come
first or the Poisson brackets come first
it does matter the order that you write
down the Poisson bracket itself that
order does matter because you have to
change the sign if you were to change
them yeah scalar has to do with their
properties under rotation of space and
so they could be components of vectors
vectors meaning
oh-oh-oh we're not talking about matrix
multiplication if that's what you mean
we're not talking about vector we're
talking about just numerical
multiplication yeah let's not call them
scalars let's call them numbers let's
call them numbers numbers which which
simply mean that when you multiply two
numbers it doesn't matter which order
you multiply them oh do they now
physicists use the word scalar to
represent the property of a system under
rotation we're going to get into
linguistic ambiguities here because a
because it may be that mathematicians
use the terminology different than
physicists business just means by a
scalar something which doesn't transform
in the rotation of space okay ah
mathematicians may use the word scaler
to mean a number physicists often use
the word see number meaning a constant
number that it doesn't matter whether
it's constant or not this is a whole
terminology here that will get confused
about but the important the important
thing is we're talking about
multiplications which are commutative
but the Poisson bracket process itself
is not commutative
eight times be eight Poisson bracket
with P is not be pass on bracket with a
okay I think we know what we mean those
are the basic abstract rules and then
there's some concrete rules if the
functions a and B are simply P's and Q's
and nothing but P's and Q's then we
presume that there are some simple laws
of Poisson brackets or simple rules per
Poisson brackets for peas with Q's peas
with P's and Q's with Q's
ah if we only had one P and one Q then
we could immediately write down that P
with P with now means plus own bracket
width I get tired of saying take the
Poisson bracket with with simply means
take a plus on break
P with P can only be equal to zero that
follows from the anti symmetry P with P
is equal to minus P with PE from the top
equation if we interchange the order but
if something is equal to its own
negative it's got to be equal to zero
same is true with cube hmm
what's that right but that doesn't tell
you what the Poisson bracket of P with Q
is let's take Q with P for that we need
to use the definition of the Poisson
bracket the definition in terms of
derivatives and then we just find out
that it's equal to one yeah now if we
put if we put subscript in we can't be
quite as confident that this is true for
every I and J nevertheless by the
definition of Poisson brackets in terms
of derivatives remember that every
Poisson bracket has in it derivatives
with respect to P and derivatives with
respect to Q well you can't
differentiate a P with respect to Q you
get zero so anytime you take a Poisson
bracket of a P with the P you must get 0
or Q where the kill you must get zero
but a Q with a P you get 1 or 0
depending on whether I is equal to J
Delta IJ delta IJ the Kronecker symbol
chroniker chronica
I don't know if I spelled that right or
not look good no see here I don't know
yeah see the zero or one depending on
whether I is equal to J or not namely
it's equal to 1 if I is equal to J and
zero otherwise
that's the chronica Delta symbol and
that's the these are the Poisson bracket
relations for P and Q now it's easy to
prove that if you take well in fact you
can prove you can prove almost
everything about Poisson brackets just
from these relationships but if you also
allow yourself to use the definition of
the Poisson bracket which can be derived
incidentally from these relations but if
you use that you can also immediately
prove that the Poisson bracket of any
function of the Q's not the peas with
any other function of the Q's in other
words if both functions only have Q in
them that's automatically 0 why because
again every Poisson bracket will have
derivatives with respect to P and
derivatives with respect to Q in its
definition so at least so it will turn
out to be 0 if there's nothing to
differentiate with respect to P so to
speak all right so this will be 0
likewise any function of P with any
other function of P will also be equal
to 0 it's only when you take Plus own
brackets between functions of P and with
functions in which at least one of them
has both P and Q in no that's not right
what do we have to say something on one
side has to mismatch something on the
other side there's got to be a p on one
side and a Q on the other side to get
anything nonzero okay so
that's a collection of rules that there
are more rules but they all follow from
these there's no independent rules for
example supposing we take the Poisson
bracket of any function of Q with P
Poisson bracket of any function of Q
with P what is that equal to well I'll
tell you right now what it's equal to
and then we'll try to prove it that is
equal to the derivative of F with
respect to Q so taking a Poisson bracket
with P is the same as differentiating
with respect to Q we can see that in a
number of ways one way of seeing it is
just use the definition but another way
of seeing it this here we can prove it
by mathematical induction let's prove it
for all polynomials F of Q we prove
something like this for every
conceivable polynomial it will be true
for every function every function can be
arbitrarily well approximated by a
sufficiently complicated polynomial are
the linearity would tell us we need to
only prove it from monomials so let's
prove this for some monomials let's take
one let's take f of Q equals one what
does this equation say for a fifth Q
equals 1 it says the Poisson bracket of
1 1 is a function it's a perfectly good
function it doesn't vary but it but it
is a function one with P must be equal
to the derivative of 1 with respect to Q
what is the derivative of 1 with respect
to Q 0 now how do I prove this I'll tell
you how I can prove it hmm
ya-ya-ya-ya-ya if i improve these but
let me give you a simple proof of this
one a simple proof of this one is to use
the product formula one squared with P
is equal to 1 with 1 P plus 1 P with 1
which is equal to twice plus on bracket
of 1 with P but 1 squared is just 1 so
this says that the Poisson bracket of 1
with P is twice the Poisson bracket of 1
with P that proves it zero ok so indeed
the first one is right let's go to the
next monomial Q with P that's the next
more complicated monomial for F Q with P
should be equal to the derivative of Q
with respect to Q which is just equal to
1 but that's certainly true that is the
defining property of Q with P let's just
go one more Q squared with P if u
squared with P for that we use the
product formula again and that gives us
Q with Q P plus Q P with Q plus some
bracket if Q with P is just 1 so this
just gives us 2 Q and that happens to be
the derivative of Q squared with respect
to Q that's a boil process of course we
use mathematical induction to prove it
for Q to the enth once you prove
difficuit to the enth then you use
linearity to prove it for every
polynomial and then the goal from
polynomials to arbitrary differentiable
smooth functions you
can use the fact that any smooth
function can be approximated arbitrarily
well by a polynomial so in fact we see
that Poisson bracketing with P is simply
another way to differentiate likewise
differentiating with respect to P is the
same as taking the Poisson bracket with
Q
notice there's a slight difference in
the order of plus on brackets here to
differentiate with respect to Q you take
the Poisson bracket with P with P on the
right hand side to differentiate with
respect to P you um you put the Q first
alright so that's another property yeah
yeah
assuming everything is continuous
functions every function is continuous
differentiable multiplied differentiable
and we're swimming that yeah
all right so we have our you could say
we have an algebra of Poisson brackets
that characterize the relationships
between P's and Q's in the phase space
now we add to that another postulate
about the time evolution of a system and
that postulate again can be derived it
can be completely derived from
Hamilton's equations we've derived it
we've derived that at least twice as I
remember but let's just write it down it
can be just added for these postulates
is one more postulate and it is that the
time derivative a dot of any function of
P and Q let's just remember what that
means we have a phase space which is a
foam which is coordinate eyes by P's and
Q's we have points of phase space moving
through the phase space under the
equations of motion of a system and as
they move through the phase space the
value of a of P and Q will change along
the orbit of the phase point that's what
a dot means the rate of change of a of P
and Q because you're moving along the
orbit in phase space that time
derivative is the Poisson bracket of a
with H this basically sets the pattern
that a flow a flow applied to a function
where a function on the phase space and
we apply the flow to it I don't reply
the flow to it that causes the function
to change as we move along the flow that
flow is characterized as a Poisson
bracket with something that something
happens to be the Hamiltonian if we're
talking about the genuine flow with time
okay in other words the real flow of the
system as time evolves
but we may just be interested in some
mathematical flow that has nothing to do
with the time evolution but just has to
do with the fact for example that we're
studying some symmetry which might
correspond to to a change of coordinates
through a sequence of intermediate the
coordinates of coordinate configurations
yeah yeah yeah yeah but of course we
could apply yes but we could just apply
this to the P's and Q's and see that
they're functions of time just by saying
the p dot p that is plus on bracket of p
with h and then if we know h if we know
what H is as a function of P and Q let's
for example let's just suppose let's
just see what happens if our Hamiltonian
happens to be P squared over 2m and we
apply we apply this to P dot it tells us
the p dot time derivative of P is the
Poisson bracket of p with P squared over
2m well that's 0 because of a formula
which I think I have erased which says
that the Poisson bracket of any function
of p with any other function of p is
zero ok so that's zero that tells us
momentum conservation on the other hand
Q dot is Q with P now we use the product
formula we use the product formula we
have P squared here and Q here the
product formula can be used to tell us
that this that this is just P over m
we have P squared over 2m that gives us
a P with the Q is 1 well you know how to
do this just gives us a P over m without
ever using anything other than the
algebra of these functions of the
algebra of these Poisson brackets we can
conclude that the Poisson bracket of Q
with P squared over 2m is P over m and
that tells us that Q dot excuse me that
Q dot is P over m and that of course is
the velocity as a function of momentum
for a non relativistic particle with
Hamiltonian P squared over 2m all right
so without knowing Hamilton's equations
we could derive the equations of
Hamilton's equations from the algebra of
these Poisson brackets okay now let's
come to a wider class of transformations
yes hmm
yep but it's a little bit more
complicated yes it does these basic
formulas hold for every physical system
that I know of I don't know of any
physical system where these basic
formulas don't hold including general
relativity including quantum field
theory well in the appropriate form in
quantum mechanical form in quantum
mechanics a whole but for all classical
C jewelr that's all classical systems
these formulas absolutely hold quantum
fields oh sorry field theory
electromagnetism everything
okay now let's move on to symmetries
what are symmetries symmetries are
coordinate our transformations
transformations of a system which don't
change the dynamics now in the cases
we've studied up till now symmetries
were typically of the form changes of
the variables Q without mixing in the
P's for example rotations of space
mix-up the X's and the Y's they don't
mix in the momenta with X's and Y's
could there be more interesting
symmetries of nature which do mix the
P's and Q's and if they do what are the
rules and what are the laws which allow
you to mix up P's and Q's and preserve
the structure of classical mechanics
that's the question what are the
allowable transformations that don't
screw up mechanics so badly that it
becomes unrecognizable recognizable
means that it has the Poisson form so
that's the first question we want to ask
what are the allowable transformations
on phase space allowable transformations
mean they don't ruin this poisson
structure that we've written down if you
were just to take a bunch of P's and Q's
which satisfy this algebra of Poisson
brackets and make arbitrary
transformations let's let's do it this
way let's
let's suppose we have a single P and a
single Q now let me make a
transformation I'm going to make a new
transformation two new P's and Q's
capital P which is going to be twice
little P and capital Q which is twice
little Q does this preserve the Poisson
bracket structure it does not let me
show you why let's calculate the Poisson
bracket of big P with big Q well that's
easy I don't even have to do any work
here I just know that this is the
Poisson bracket of twice little P with
twice little Q what's that for
it's the wrong answer it does not
preserve the form of the Poisson bracket
relations let me try something else
instead of big p equals twice little p
let's take it to be one-half little p
now what happens
what's the Poisson bracket of big P with
a big Q one this formula has preserved
the structure of Poisson brackets if I
do this transformation on the P's and
Q's it's not necessarily a symmetry of a
physical system but it is another set of
coordinates which has exactly the same
Poisson bracket relations as the
original P's and Q's kind of interesting
well I've done enough to the phase space
is I have squeezed it in one direction
by a factor of two and stretched it in
the other direction by a factor of two
and that has preserved the structure of
the Poisson bracket relations stretching
both directions by a factor of two did
not
serve the relations how about the
following supposing I let's let's try it
another example let's take okay it's
basically conservation of area yeah but
it's conservation it's it's a little
more complicated once you leave the
domain of just one P in one Q it's more
complicated but for one P and one Q the
the transformations that we're talking
about are the ones that preserve areas
for more peas and molecules it's more
complicated but let's just do one or one
or two more examples um let's take big P
is let's rotate P into Q rotate P into Q
in other words we have the PQ axes and
we rotate let's take big P equals cosine
theta little P plus sine theta little Q
and let's take big Q equal minus sine of
theta little P plus cosine theta little
Q this is the form of a rotation in
phase space it's a rotation in phase
space where you've rotated the axes from
what I just told you about areas it's
quite clear that this also should
preserve the structure of Poisson
brackets let's check it let's calculate
the plus own bracket of big P with big Q
do I have to check the Poisson bracket a
big p with big P big P with big P this
is linguistically the hardest lecture
for me though I have to check the
Poisson bracket a big P with too many
B's and peas big P with big P
what's that well it's just a big P with
big P antisymmetry tells me zero big P
with big P has to be the same as big P
with big P in the opposite order that
tells me to zero antisymmetry tells me
big P with big P is zero it tells me big
Q with big Q is zero it's only big P
with big Q that I got to watch out for
okay so let's just do it so that's going
to be ah let's big big Q with big P
equals cosine sorry sine the sine theta
now let's let's begin with kills all
right cosine theta Q plus sine theta P
that's this one here there's one of them
- one of them is - right this one is -
and then the other one is cosine theta P
plus sine theta kill all right the only
combinations which give me anything are
a Q of the P and a P with a Q let's
first take the Q with the P over here
that's going to have cosine squared
theta in front of it
cosine squared theta plus on bracket of
Q with P what's on bracket of Q with P
is 1 and then we're going to have minus
sine squared theta that's coming from
here what's the Poisson bracket of P
with kill minus 1 so that corrects this
sign here and makes it plus sine squared
and gives us a 1 again all right so the
rotation of coordinates preserve the
Poisson bracket structure the stretching
of 1 coordinate and the
weezing of the other coordinate by the
same factor it didn't have to be too
incidentally that will preserve the
Poisson bracket structure on the other
hand other kinds of transformations will
not preserve it in particular those that
don't preserve the area so there are
those which do in those which don't
those transformations which preserve the
entire class of them and it can be much
more complicated than this in general
for arbitrary numbers of P's and Q's the
transformations which preserve the
Poisson bracket structure are called
canonical transformations canonical
canonical transformations are those
which keep intact the relations between
probably the Poisson bracket relations
between P's and Q's if they keep the
relation between P's and Q's then they
will keep all the Poisson bracket
relations between P Square Q squared P
whatever you have what what's that yeah
but you have to remember that when you
change that the Hamiltonian has to be
expressed in terms of the new variables
yes that's correct
that's right that's right these are not
these are not gauge transformations are
now these are canonical transformations
they're not gauge transformations the
not redundancies of a description it
just changes of the description to new
coordinates
right farming the quarter well those are
the Colgate's transformations
each transformations have a definite
meaning now points if you start with
canonical pees and kills and you do
canonical transformations on them the
they stay canonical now that's a
tautology but to make it not look at the
tautology what do you mean by canonical
you mean these Poisson bracket relations
these are the canonical Poisson bracket
relations if after transformation all of
the Poisson bracket relations are kept
intact then it's called the canonical
transformation I guess what's a
we're big peasant big kids being
functions of the small peas and smokers
there's changes of coordinates is what
they are the changes of PQ coordinates
no no no no no no no we're talking about
different coordinates for the same
system that's all we're talking about
different coordinates for the same
system and by coordinates I now mean
coordinates on phase space so we're
talking about different representations
of the same system by different choices
of axes or different choices of
coordinates on the same phase space no
no no no no we're talking about the same
system yeah these are changes of
description rather than changes of
system okay changes of description of a
given system right we must keep that in
mind it doesn't mean anything for
example to to change the coordinates of
from a harmonic oscillator to a system
of 10 10 to the 23rd molecules it
doesn't mean anything
it doesn't mean anything I mean if we if
it does mean something we have to make
up a definition for what it means here
we thought we're talking about P's and
Q's which are really functions of the
original P's and Q's which means that
just read descriptions of the same
system yeah
No
the phase space of end of n degrees of
freedom are all isomorphic to each other
but still you don't you don't
nevertheless when you're talking about
canonical transformations you're talking
about the changes of description of a
given system yeah but the different
phase spaces all with the same number of
degrees of freedom are isomorphic to
each other and that's correct yeah yeah
but still you don't transform between
systems right all right now we want to
come towards the heart of the discussion
the subject of symmetries itself
um this is the necessary and sufficient
condition that the p's and q's preserve
the comma there they're Poisson bracket
structure that is the necessary now
you're asking you're asking if you give
me a certain well what are you asking
yeah yes yes there is another way of
expressing it that is of great general
interest but is also not completely
general we build that we tend to build
transformations not all of them but we
tend to build transformations out of
infinitesimal transformations out of
small steps okay if we're talking about
an infinitesimal transformation I'll
explain exactly what I mean by an
infinitesimal transformation in the
moment but I think you know what I mean
though yeah yeah if we're talking about
infinitesimal transformations then yes
there is a way to describe them a unique
special way to describe them which is
which we're about to come to now so if
we can build up the transformations out
of infinitesimal transformations and
what is an infinitesimal transformation
it means one well it's just write down
what it means
it means Q is almost exactly equal to
little Q but with a very small deviation
and that small deviation will just call
Delta sub Q Delta of Q and this of
course is a function of the Q's and the
peas it just means the deviation from a
trivial transformation big Q equals
little Q is infinitesimal
in practice what it means is in their
formulas we will drop things
proportional to quadratic and higher
powers of the deltas it means that small
quantities mean quantities which are so
small that we can drop higher powers in
them that's what will always mean by an
infinitesimal operation that we can drop
higher powers now we can always build
transformations out of sequences of
little transformations for each one of
which we can drop higher powers but then
we have to combine them together and
sometimes that make something more
complicated but we can test out very
easily what are the rules for
infinitesimal canonical transformations
I won't bother writing down that it's a
function of P and Q all right let's just
remember that the deltas are themselves
functions they're functions but they're
small such that we can ignore higher
powers of them
alright let's first write down the
condition that this be a canonical
transformation incidentally of course
there are some indices here I'll
suppress the indices up until the last
step let's write down the condition that
this be a canonical transformation it
just says that all right so let's just
do it for let's just do it as if it was
just one P in one Q and we can fill in
the gaps later okay um
Poisson bracket a big P with big Q
that's the interesting one here big P
with big Q that's just going to be the
Poisson bracket a little Q with little
PE will stick in the eyes
later plus the plus own bracket of Delta
Q with little P plus the Poisson bracket
of Q with Delta P I've just stuck in for
big q q plus Delta Q for big P P plus
Delta P and there's in then use
linearity to expand it out I've left out
one term I've left out the Poisson
bracket of Delta Q with Delta P but
that's already quadratic in small
quantities quadratic and small
quantities are too small to be
significant so this is our formula now
what do we want to be true we want this
to be equal to this so necessary and
sufficient condition then is that Delta
Q with P should be equal to minus Q with
Delta P let me show you how to satisfy
that this actually is necessary and
sufficient but it's it's not hard to
show that it's necessary and sufficient
but let me just show you the
construction what do we do
we write that Delta Q is equal to some
small number this this the epsilon is
just a small number in the problem Delta
Q Delta PR all is proportional to the
same small number I introduced
explicitly the the small number so that
I'll know so I keep track of it times
the Poisson bracket of Q with something
with what with an object that's called
the generator of a canonical
transformation G of P and Q and Delta P
is equal to epsilon times P with G
another way to say it is that every
cannot every infinitesimally canonical
transformation is characterized by a
function G of P and Q which itself
defines a flow same order same order
this is a kind of flow according the
transformation in particular an
infinitesimal coordinate transformation
there's a kind of flow on the phase
space you take every point and you shift
it a little bit you take every point
then you shift it a little bit that's a
flow on the phase space and flows on the
phase space are characterized by things
called generators generators are
functions of P and Q and they
characterize how the flow takes place
let me prove now the theorem will prove
the theorem that says that if Delta Q
and Delta P are obtained by Poisson
brackets with a generator then this will
always be true it goes the other way
also this is uh this is not hard to
prove first of all what is the Poisson
bracket of Q with G of P and Q remember
taking a Poisson bracket with Q is the
same as differentiating with respect to
P
what is this is DQ by DQ DG by D P -
yeah this is just epsilon the G by D P
just using the definition of Poisson
bracket now the Q by DQ DT by DP that
just gives you the G by DP and then the
Q by DP at at zero what about this one
this one is equal to minus epsilon the G
by DQ so now we have specific formulas
for Delta P and Delta Q here they are
let's plug them into this formula over
here what is Delta Q Delta Q is epsilon
VG by DP epsilon plus on bracket of the
G by DP with P and that should equal
minus plus on bracket of Q with Delta P
what's what's Delta P again Delta P is
minus DG by D Q right
Epsilon all right the minus signs cancel
themselves out I hope and what is the
Poisson bracket with P it just
differentiates with respect to Q so this
is just the second derivative of G with
respect to P and Q what's Poisson
bracket with Q derivative with respect
to P this is identically the left side
and the right side are identically equal
to each other
they're identically equal to each other
each one being the second derivative of
G with respect to P and Q so okay so
here's what we find we find that flows
which are generated by Poisson
bracketing with respect to a generator
always define canonical transformations
a special case of this is the
Hamiltonian flow let's just look at this
equation here what does it say supposing
we were talking about the time motion of
a system the time motion of a system is
Q dot equals Poisson bracket of Q with H
well if I write the Q dot is Delta Q by
delta T and I just think of delta T as a
small epsilon but epsilon here in other
words I change the time interval by
amount Epsilon then the Hamiltonian flow
is exactly the same as the sort of thing
I've written here Hamiltonian evolution
of a system is a canonical
transformation in other words the
coordinate transformations of P's and
Q's which are generated by the actual
flow of the system is itself a special
case of a canonical transformation
furthermore all Kanaan
transformations can be generated by
picking some generator as if it were the
hamiltonian imagining in our mind a
Hamiltonian which caused the flow which
took the points from one to another that
defines the most general class of
canonical transformations now yeah
the most general I think I'm saying
something a little too strong I'm not
sure maybe I'm not on every canonical
transformation that I know now I know
I'm saying something correct every
canonical transformation that I know of
I think it's true can be generated from
infinitesimal canonical transformations
if it can be generated from if it can be
generated from an infinitesimal
canonical transformation then it's got a
generator G of P and Q which are which
allows it to be constructed in this way
so if you want a small canonical
transformation which just displaces the
points a little bit you just take
Poisson brackets of the variables you're
interested in with the generator and
that gives you the changes in the in the
phase space coordinates that's the basic
idea that all motions on the phase space
whether they have to do with the actual
time evolution or whether they just have
to do with coordinate transformations
moving points around are of the form of
canonical transformations generated by
generators the Hamiltonian being a
special case yeah
yeah yeah yeah yes it's just it could be
the Hamiltonian for another system then
the Hamiltonian could just be some
arbitrarily arbitrary flow ah that's
right what the Hamiltonian is of course
is really determined by the physics of a
system experiment and all that kind of
thing but the mathematics cannot tell
you whether G is the Hamiltonian or H is
the Hamiltonian you do an experiment to
find out okay now we can now we can
discuss if the transformations which we
are allowed to do which preserve the
structure mechanics are canonical then
what is the subclass of roles for a
given system with a given Hamiltonian
what is the subclass that are called
symmetries symmetries are the canonical
transformations which don't change the
Hamiltonian just like in the case of
lagrangians we said that the symmetries
were the coordinate transformations
which didn't change the Lagrangian now
the more general idea becomes coordinate
transformations in phase space which are
canonical and which don't change the
energy those are the symmetries of a
mechanical system we can actually say
that in a more in a somewhat geometric
way here's our phase space Oh
incidentally I didn't keep track of the
little indices I you can go home and do
the derivation over again keeping track
of all the indices I and J and see that
the construction still works that's it's
more or less obvious yeah okay but let's
imagine he is off a space multi
dimensional phase space now the
Hamiltonian corresponds to some flow and
it's also a canonical transformation the
infinitesimal small time flow from one
point to another is the canonical trends
as the is generated by the Hamilton
I mean and the full transformation over
a period of time is canonical now let's
imagine that some other quantity a
generator G that generates another flow
it's not to be thought of as the time
the time evolution of the system is just
another flow on the phase space just
another motion on the phase space which
moves points through the phase space in
some other way in general you can see
from this picture that are the new flow
the green flow will change the value of
the energy along the flow but supposing
it's true supposing it's true that the
green flow G and the black flow H the
Hamiltonian of the system are in just
such a way that the flow along the Z
Direction doesn't change the energy
causes the energy not to change at all
in other words flows the system along
surfaces of constant energy then it's
called a symmetry it's a symmetry G is a
symmetry if the flow created by G
doesn't change the energy what's the
condition for that this is easy hmm
right right geometrically the condition
but in a multi-dimensional space it's
not so obvious all right um here's how Q
and P change under the flow let's take a
general function of P and Q let's take a
general function of P and Q and see how
it changes along this flow
I'm not going to call it I'm going to
call a function a again let's call it a
how does a change along the flow well it
changes the a by D P times Delta P plus
PA by DQ Delta Q but what is Delta P and
Delta Q here we have them they're
proportional VP but the G by DP and DG
by DQ so this is let's see Delta P is
the G by DQ the G by VP in other words
not too surprisingly the change in any
function of a is just proportional as an
epsilon and here isn't there it's just
epsilon times the Poisson bracket of a
with G this is exactly the same formula
we use for time derivatives of things
when G was the Hamiltonian the change of
functions along the Hamiltonian flow is
just a Poisson bracket with the
Hamiltonian the change of an arbitrary
function along any flow is just a
Poisson bracket with the generator of a
flow that's what these generators do
Poisson brackets with the generators
create the flow and they give you the
derivatives or the way the functions
change along the flow alright so then
what does it say that the energy does
not change along the flow G it says that
the Poisson bracket
of H with G is equal to zero it says
that H does not change along the flow G
right if I were to just plug in here for
a e
H everywhere is this would just be the
Poisson bracket of H with G so this is
the condition that the energy doesn't
change along the symmetry flow G but now
we can read this the other way also this
also tells us that G with H is equal to
zero H with jeans you with H are just
the negative of each other and what does
this equation say this says that the
time derivative of G vanishes right
remember the other equation was that a
with H equals a dot the same equation
which says that H doesn't vary along the
along the flow G says the G doesn't flow
very along the flow H but what does this
equation say it says that a is conserved
it says that a is conserved so if you
have a Hamiltonian and you have a
generator G a quantity a quantity G
which has the property that the flows
created by G don't change the energy
then the flows created by the energy
that's the time dependence of the system
don't change G
this is kind of the reciprocal relation
is an interesting reciprocal
relationship if you have two quantities
H and G and you say H is the Hamiltonian
which generates the time dependence and
G doesn't change the Hamiltonian then if
you switch the system and you said G was
the Hamiltonian and H was just some
arbitrary thing I think you get the
point I'm going to I'm going to use too
many words to say it there's a parallel
relationship between quantities which
are conserved and the energy of a system
parallel symmetrical relationship
between them I think this would be a
good point to stop this is a very very
highly abstract business and it does
have to do with the intrinsic structure
of classical mechanics but as I said to
you climbing again the place where the
real glory of this comes forth is in
quantum mechanics it's very abstract in
classical mechanics it's extremely
concrete in in quantum mechanics much
easier to understand in quantum
mechanics of what angular momentum and
energy H could be let's see H could be
the energy of a system it could be P
squared over 2m plus I don't know
function of a function of X square plus
y squared let's say that could be H G
could be the angular momentum angular
momentum is X py minus YP X if you
calculate the Poisson bracket between
this and this you find that at 0
corresponding to let's do that oh yeah
it's going to be complicated let's do it
for a free particle it's going to be a
little too complicated to do in general
yeah let's do it for a free particle ok
so let's take P squared over 2m and that
means px square plus py squared
you mind if I set em equal to one now
pay x squared over 2 plus P y squared
over 2 that's the Hamiltonian G is the
angular momentum the expression for the
end this is two dimensions just for
simplicity two dimensions G is the
angular momentum that's xpy minus ypx
okay now we want to start calculating
Poisson brackets between G and H this is
not too bad
remember all peas hmm
no this isn't peas right yep okay X's
and Y's are Q's good all right it
wouldn't be true if I just took X py or
YP X but it's a combination X py minus
one okay so let's do it plus own bracket
of G with H okay so what is that it's
going to have X py with px and it's
going to have X py with py
how about let's do for X P Y with py
first of all do I get anything from that
the Poisson bracket of an X for the py
is zero remember the only nonzero
Poisson brackets that you begin with our
X with P X & Y with py when I start
calculating X P Y with py I'm not going
to get anything py plus 1 bracket with P
y is 0 X with P Y is 0 so there's
nothing on the side which has a non zero
plus own bracket with anything over here
the only non zero Poisson brackets are X
with P X & Y with py okay so let's begin
with X with px
let's this is going to be them Poisson
bracket of X P Y with px squared over
two and then the other term will be
minus y PX with py squared over two with
py squared over two
okay how about X py with px squared over
two well we use the product formula here
we take out one term is just py with XP
x squared over two then there's another
term where the X comes out and py gets
plus owned with px but what's plus sone
of X P Y with px 0 so this is the only
term coming from the first bracket py x
X with px squared
what's X with P X squared remember what
Poisson bracket with a Q does it
differentiates so X with px squared
gives what just px this whole thing is
just P YP X it's just py px what about
this one we'll get a px on the outside
and we'll get Y with py squared which
will be py which will just be minus py P
X so this is an example of two
generators which have zero I keep
wanting to use the word commute and
quantum mechanics it would be commute
but let's not use that terminology this
is an example of two quantities two
generators which have vanishing Poisson
bracket between them the implication is
the following if the Hamiltonian is px
square plus py squared then the angular
momentum is conserved
if we invented a new system where the
Hamiltonian happens to be this thing
crazy Hamiltonian but let's do it anyway
a new system where the Hamiltonian is
this thing then this object would be
conserved the Poisson bracket between
them is zero so whichever one is the
Hamiltonian the other one is conserved
and again the idea that the Hamiltonian
doesn't change under a transformation
that is the basic idea of a symmetry
that a symmetry is an operation which
doesn't change the energy but we see
here this a rich structure to the whole
thing a much richer structure then then
well then you might have expected okay
any of any questions about this and then
I want to move on to I want to move back
and review what we said the first time
yeah what said yeah Oh
let's see what it means let's see what
it means to see what it means it means
it simply means let's find out the
equations of motion that go with it and
ask how a particle would move under the
influence of that Hamiltonian know a
thing and then work it out here's a
Hamiltonian new system and meltonian is
X py minus YP X let's work out
Hamilton's equations Hamilton's
equations say X dot is equal to the
Poisson bracket of X with H what's the
Poisson bracket of X with H can anybody
read this off where does it come from
which term the only terminates which can
contribute to a Poisson bracket with X
is a term that has P X in it so this
looks like it's going to be minus y X
dot equals minus y and what about Y dot
why that is equal to X anybody have any
idea what kind of motion this is circle
circular motion it just describes
something going in a circle yeah I meant
to change now so let's make up a new
system where the Hamiltonian just
happens to be X py minus YP X what do I
get I get a system which just moves in a
circle counterclockwise direction like
so so if we had some mechanical system
which could only move in a circle we
would describe it by this Hamiltonian
but then we would find that px square
plus py squared was conserved and in
fact that would be true
you've gone through these with stupi in
the queue and myself I want confirms my
senses you're always talking about the
same index well for each yes yeah the
transformation from each index the
transformation could be different yes
yeah yeah but it's more complicated than
there on the transformation no there's
one there okay it's not that there's a
transformation for each index there's a
set of transformations there is a
transformation of reaching the exit
ticket back Qi is a function not just of
Qi and P I but of all the Q's and PS
okay so there's a transformation for
each index but each transformation in
general involves all accused Peas yeah
right yeah
for you say it again
Oh how do you get the momentum in terms
of queues um I think for this system py
is actually equal to X and px is equal
to minus y let's see can we see why that
is
you mean how do we find the peas in
terms of the velocities I think the I
think the question you're asking is how
do you know the peas in terms of the
velocities I think that's the only
question I can think of it you could ask
it's not always MV no but you go to
Hamilton's equations you go to
Hamilton's equations and let's do it for
an ordinary case and see how it works
for an ordinary case we could have H is
equal to P squared plus plus some U of X
okay let's just do it for them excuse me
let's just do it for the good old
familiar simple particle and we'll see
the pattern all right so we just go to
Hamilton's equations Hamilton's
equations are x dot equals plus some
bracket of X with it with H and that's
just going to give us P over m that's it
that's it that tells us P in terms of X
dot the other equation is the equation
that P dot is equal to P with H and that
gives us arm D u by DX and that's of
course F equals MA force is equal to
mass times acceleration so one of the
hamilton equations typically the usual
situation is one of the hamilton
equations tells you the relation between
the peas and the x dots and the other is
like newton's equations involves the
acceleration yeah
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