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we're going to plunge deeply deeply
deeply in for the Hamiltonian mechanics
of our mechanics we've studied four
Lagrangian method we've studied the
principle of least action we've even
gone to the extremely advanced study of
Newton's equations tonight
it's Hamilton's night
part of the point that may be the main
point of the Hamiltonian formulation of
mechanics goes back to something that I
talked about at the very first lecture
we talked about the basic structure of
classical thinking classical mechanics
but not classical mechanics quite but
something even a little more primitive
we talked about discrete systems the
corner which could either be heads or
tails and that's a two-state system two
simple states and then we talked about
possible laws of evolution one simple
walls evolution is nothing happens
evolution now might need a sequence of
states separated by a small time
interval or discrete time interval apart
from a second or whatever we update
every tenth of a second then we spoke
about a simple law of nature's which
were a simple law of coin flipping a
coin changing which would be no change
at all here without doing anything
that's a simple evolution another less
simple law of evolution is if it fits
then the next time it stands the tails
the next time it's heads tails heads
tails heads tails heads both of these
rules have the feature that the two
normal the state of the system at any
instant of time you know it at the next
instant of time in other words the
updating rule is precise but also the
backdating rule it's also precise if you
know where you are at any instant you
know where you work we even talked about
the general structure of such
simple theories instead of just two
states we could imagine a system with
many space each one being represented by
a point in a space we might as well now
call that space phase space it's the
phase space of a simple discrete system
in this case it's got one two three four
five six if I stop there with only six
states that we could be talking about a
dog dies and vice six possible
configurations add two more I don't know
it could be something what sister has
eight stairs three coins three coins
precisely so this is a general kind of
situation and the basic setup in fact
the definition of a phase space is if
you know where you are in the phase
space you know where you'll be next and
you know where you came from
so the role which could be represented
simply by a set of arrows which tells
you where to go next
for example you can have other kinds of
laws not different than in kind but
different in detail for example you
could have some closed loops like this
so wherever you want you go to the next
place but now listen close cycles so
that if you're on this cycle you'll stay
there and if you're on this cycle you'll
stay there this corresponds to the
existence of a conservation law we can
imagine that this these three states
over here are labeled as having property
equal to plus one and over here property
the same property with minus one
well if you've got plus one you stay
plus one plus one never becomes minus 1
and minus one never becomes plus one
because all of the configurations with
number minus one though the
configurations with number minus one and
so forth but both of these kinds of laws
have the property if you know what you
are you know where you'll be and you
know where you were
that's to be contrasted with a law which
might look something like this if you're
over here you go to here if you're over
here you look to here and if you're over
here you go to here that's a very
definite updating room wherever you are
you know it feel next but there's
something funny about it namely in
particular you can't go backwards and be
unique in the plane in a unique way
you can go backward from this point from
this point you know that you must have
come from here but if you're over here
you don't know whether you came from
here or whether you came from here
what's going on here is points on
running into each other does yet another
way to think of it imagines the phase
space being populated populated with
points and the point attack let's
imagine the entire face
is populated with red dots and the red
dots all move they go dance the dancers
simply to follow the trajectories as
they're prescribed by the law whatever
the law is some of this red pop comes to
here this red pad posted here mr. Don
Costa here this red posted here every
red dot knows where to go knows where to
come from and there's a kind of rigidity
to the red bottle to the red dots none
disappear
none spontaneously a peer long disappear
none happier they simply just rigidly or
what's it even better word than rigidly
a fluid incompressible incompressible
motion where points don't run into each
other same thing here same thing here if
we populate the points with red dots and
we allow them to move along the arrows
instant by instant by instant each
instant the number of red dots is the
same they don't run into each other they
don't bifurcate the whole thing just
moves in an incompressible way on the
other hand with this law we could start
a red point at each place but then after
one motion this one would have
disappeared never to reappear and we
would have only two places which were
occupied going back and forth back and
forth there would be some loss in a
sense of points in the face space
they spaced points would not be
conserved
the whole point
mechanics this is the born into form in
which every point knows where to go next
and those where it came from
without points ever being lost or gained
in other words a kind of rigid motion in
a face space or incompressible motion of
the points in phase space which may also
be called information conservation
we'll see we'll talk about that a little
bit further in the future ok another
point of Hamilton's equations let's
let's think about ordinary F equals MA
equations they're not of this form
why not well let's suppose you know
where you are
let's stick there's the positions of the
particle possible positions of a
particle is the particle x y&z supposing
I know the particle is over here
do I know where will be next no I need
to know something else I need to know
not only where it is now but I need to
know where it was an instant back in
other words I need to know the position
and the velocity knowing the position
the velocity is the same thing is
knowing the position and listing and
knowing the position a slight bit
earlier so I need two pieces of
information I need to know whether the
object is now I also need a kind of
ghost piece of information which is
where it was an instant ago and that's
enough to tell me where it will be next
next means after a short prescribed
period of time so what specifying the
local
a sister by now we can simply call them
the cues of a sister does not cause the
commune a space the phase space
constitutes everything you need to know
in order to tell where you'll be next
but in such a way that is a rigidity to
the motion a rigidity to the motion
points of a gained or lost
that's what phase space is phase space
is a way of representing the
configurations of a mechanical system so
that the motions are described much as
I've described these discrete discrete
motion we'll come back to it the detail
I think the Lagrangian formulation was
formulated by people for practical
reasons the practical reasons being most
systems were awfully complicated and
difficult to apply ethical czar many and
so a more general more general
formulation which was not dependent on
particular sets of coordinates in many
many contexts rectilinear coordinates
are simply not appropriate for problem
components of vectors are not
appropriate to a problem or not
particularly useful motion for example
motion of a particle on the flat
blackboard can be nicely described by
X's and Y's and we can use x and y
components and write F equals MA motion
on the surface of a sphere
I'm not jacking around on the surface of
a sphere it's not natural
special sensitive introduced rectangular
coordinates you can't introduce
rectangular coordinates on the sphere
you can introduce coordinates latitude
and longitude but you can introduce
rectangular coordinates the naive idea
of components naive idea of components
or angular Cartesian coordinates is not
useful so it's more it's useful to
define mechanics in a way that doesn't
depend on any specific set of
coordinates that was the little that was
Lagrange's more than reservation what
Hamilton was smoking I don't know but
what he developed was an extremely
beautiful elegant formulation of
mechanics which turned out which they
don't think he ever really used for
anything he simply admired it physicist
admired and afterwards it turned out to
be very useful part of the usefulness of
it was because it does conserve
information in the same sense that I've
described over here I'm sure he was
aware of that
but I don't know if it was foremost in
his mind but ultimately the real power
of it came when quantum mechanics was
developed so it's hard for me to put
myself in Hamilton's position I mean it
was one of these brilliant mathematical
insights that kind of came from nowhere
just somehow he saw the structures there
in ways that I think most of us would
not have and so no I can't really give
you his intuition what we can do is do
it and after we've done it go out back
and talk about why it's so useful and
why power I'm not particularly sure that
anybody realized how powerful and useful
it was until call mechanics was
developed I think that really was the
time when people first really began
feeling they had to learn what happened
to say all right another way another
observation about it
is it takes a certain number of second
order differential equations F equals MA
is a second-order differential equation
for each component
second order means that it involves
second derivatives not first derivatives
the second derivatives so the males will
involve a first derivative here in there
but it contains second derivatives
it contains accelerations
second derivatives with respect to time
or something how many equations are
there well if there are a certain number
of particles in a certain number of
dimensions that tells you how many
equations there are one particle there
are three equations two particles there
are six equations and so forth each one
being a second-order differential
equation where heck works where
Hamilton's equations are in general you
can always take a system of second-order
differential equations let's say n of
them and replace them by twice as many
first-order
that only contain first derivatives it's
very easy it's so easy it's almost
stupid so let's let's discuss how do you
take in second-order equations the
method is completely general but I'll
just Illustrated for Newton's equations
supposing we have a certain number of
equations of the form derivative second
derivative of some on each component of
X or DT squared is equal the I component
of force divided by the mass I've
written acceleration is equal to F over
m I don't know why exactly
I just wanted to write the acceleration
on the left-hand side let's suppose
narrow that there are any such
differential equations let's take that
and make two n equations each of which
only contains a first time derivative to
do that first of all define now we could
do it many ways but I'm going to do it
in a particular way that in order to get
familiar with terminology
first of all let's read let's define the
X I by V T and at the end let's throw in
a factor of the Man 1 mass everybody
let's take n times the X sub I DT and
give it a name what do we call it we
call a piece of I now let's write
chemicals are made now that's right F
equals are made in the form that let's
just make one in all right now let's
take this is of course a first-order
equation it only contains first time
there it is
but now let's write f equals array in
the form DP by DT is equal to F both of
these equations are only first-order
equations but there are twice as many of
them there are twice as many equations
but each one only contains first time
derivatives they become a two in a
system of two any differential equations
but twice as many variables in our water
that more than twice as many variables X
sub I and he survived
first other equations of a kind which
tell you where to go if you know where
you are without knowing where you were a
first-order equation tells you how to
update to the next position if you know
where you are but knowing where you are
now means knowing where you are in the
phase space it's equivalent to the
overview you'll again need to know two
things a position and that's something
equivalent for velocity but it's just a
different mindset speaking about the
phase space gives you a picture XMP in
which wherever you are you know where to
go next so there's no more physics in it
normally laws of nature just a different
mindset thinking about the state of a
system and it has a configuration but as
a combination of a position and a
momentum or not just one position of
momentum but a position and a momentum
for each degree of freedom for each
coordinate more generally of course we
will talk about Q's and PS of Q's and
pies I think I'm going to give up the
notation PI but those are the various
motivations various properties of the
hamiltonian description it's the
equation of this hamiltonian description
of mechanics is the equations for how
things move around on a phase space the
only where you are they tell you where
you move next in fact you can almost
think of it just as
we before are a law as being a kind of
flow a flow of the points in phase space
we're going to see that there's a
natural sense in which mechanics
constitutes a flow in the phase space
wherever you are you know where to go
next and if you were to imagine of
populating the phase space populating
the phase space would mean taking a lot
of identical system and starting each
one at a different point in the phase
space and watching them as they evolve
that would define a flow in the space
basis it's not flow in the phase space
the Hamiltonian mechanics is all about
it's almost as if the points in phase
space were kind of fluid and we're
studying how that fluid moves if we're
interested in any specific system then
we think of it as a little particle
moving along with flow we started
someplace we allow it to move along with
the flow and follow its coordinates
momentum
all right now we need a small piece of
mathematics it's an easy piece of
mathematics though it's first in the
simplest possible case suppose there are
two variables let me call them V and P
just secretly between you and me V
stands for velocity and P stands for
momentum but now forget I forgive that
because we're going to think about the
Legendre transformation as a
mathematical trick in many many
different contexts all right so I had
there are two variables
and let's suppose but they're not
independent variables in fact is really
only one variable they're functions of
each other the functions of each other
and for simplicity not for simplicity
this is important let's suppose there's
single valued functions of each other
single valued function means that if you
know P you know V and if you know be you
know P no ambiguity that means the graph
of P versus V looks something like that
what it can't do is it's not allowed to
do that
why not because if it did this still
would be certain values of P where you
would have an ambiguity about what B is
it's also not allowed to do this because
then there would be certain values of V
where you wouldn't know what he was
single valued means that the graph of
the function P versus V looks like this
and each point on it or for any value of
B there's a unique P and for any value
of P is a unique P actually just for
simplicity although this is not
important
let's just draw the graph so that it
goes through the origin
then if we have such a single valued
function not just a single valued
function but a one-to-one function
meaning it's single valued whether you
think about it as P as a function of V
or V as a function of P then it's
possible to invent a pair of functions
one which I'll call arrow secretly the
Lagrangian the Lagrangian is a function
of V and it has the property that the
derivative of the Lagrangian with
respect to V defines P
while I bother inventing a function and
whose derivative with respect to V is
people well comfortable I but I can
always do it this is just ordinary
derivative for the moment this is only
one variable V we don't have to make it
a partial derivative D given L we can
construct a symmetrically related
function that we're going to call H it
just to be regarded as a function of P
where we think about L we will think
about V as the independent variable when
we think about H we will think about P
is the independent variable of course it
is also a function of P because V is a
function of P of course H is also a
function of V because P is a function of
V so but each one is followed as a
function of its own variable L of V or H
of P and the relationship is H and L are
connected by the derivative of H with
respect to P is V this is a completely
symmetrical relation between T and V and
between l and h the derivative with
respect to V of L is P the derivative of
P H is V so let me show you why that's
always possible to do and to show you
the relationship between L and H okay
let's take our function which the
simplicity that just had pass through
the origin but you can work it out
whether it passes through the origin
home I promise the derivative of L with
respect to V is equal to P think of that
as P as a function of V of course V is
the independent variable in the
population so P is really a function of
V as it is over here V of V P is a
function of you know just reason already
on P we can also think of V as a
function of T and then let's solve this
equation for L by just writing that L is
people to be integral of a derivative of
mammal is P then L is the integral of
people we can write this as deer does
everybody recognize what I've done
okay that can be taken to be a definite
integral with definiteness let's take it
to be the integral from zero but it
doesn't matter it doesn't matter where
we pick neither grew from but let's for
definite this take it from zero from the
equals zero if we differentiate an
integral we just get the Instagram so we
differentiate L with respect to V Aviv
in you if you differentiate the integral
with respect to the endpoint of
integration you just get the integrand
so that's this relationship here
derivative is equal to we can all you
always do this the air cuts L of the end
point if you like all right we can now
draw a picture of what L Amina's at
least on this graph L is the area L of V
let's say lob is constructed in the
following way you go to your point B
draw that line and then you consider all
the area underneath the curve here this
is not a triangle it's a curve that's
the integral of V of V V V and it's the
locus L at Point P
it has the property that you
differentiate it with respect to B you
get people now what about H well we can
do the same thing with age H must be the
integral of V DP just by symmetry your
question is that H of P is the integral
we can get started zero and go to P of V
of P VP notice if we differentiate H
with respect to P we just get B that's
the other defining equation here
derivative of H with respect to P is
equal to V alright so the geometry of
that then is that H the area in here is
H H of P and the area in here is L of B
this is this area in here is the
integral of V of P of V of P P P
a further explanation well now we know
the relationship between inch and he'll
take everything H and Antonella h plus L
is the area of this whole rectangle
what's this B this is usually written as
H is equal to PV minus L that's clearly
the same thing for some reason we tend
to write to think L first and then
trying to figure out what H is
and so now let's write our system in
equations
they're called key Legendre
transformation you start going elevate
you differentiate elevator
the fines people then gives you P as a
function of B then you construct H by
multiplying P times B and subtracting L
now L of course is a function of V but
wherever you see V substitute its value
in terms of P B but B is a function of P
so P is also a function of P here so H
gets to be thrown up as a bunch of a P
lol is a function of V and when we do
this we will not prove I'm not prove
although it's this clear from the
geometry yeah that the derivative of H
with respect to P is V let's do the
little manipulation algebraically is
what we do we consider a small change of
P and that's how much each changes how
much does a change when we change people
let's work it out the change in H let's
just call it up a small change in H is
first of all what's the change in a
product if we change that we make a
small change of the values of some
variables
what's the change in a product sum of
two times one term being the first
factor times the change in a second and
vice versa
all right so first of all we get a term
which is P times the small change in V
the second term is V what's the small
change
and finally - the derivative of the
Lagrangian derivative of L with respect
to feed
remember that L depends only on these
times the change in V
oh wait a minute I defined the elbaite
began to be called P that was the way
that I constructed P the elbow by EB is
equal to P so what might really have the
LTV I'm sorry sometimes rank partial
derivatives and sometimes total it
doesn't matter there's only one variable
partial is Pokemon the same alright so
this thing is P well here I have PPV
here I have minus PD V so these two
cancel completely store the same and
that just leaves to change it H is the
kinds of change in P that's the same
thing as saying that the H by V P is
equal to V
the H by T P is equal to V in other
words the change in H when I change P a
little bit it's just people to be okay
that is the Legendre transformation so L
to find the LDV to be P then construct H
which is PV minus L and now you have a
reciprocal relationship where the h by
DP is equal to V so if we need to change
L and we have to change that ln h and we
also have to change being v it's
completely symmetric they're completely
symmetrically related to each other this
little demonstration over here is
nothing but that's a mathematical
construction which is not only which is
at the heart of many many things in
particular mechanics but also film open
areas thermodynamics is just
transformation related in pairs now what
happens if there are many of these
instead of just one v supposing there
are many of them then the definitions
going follows the derivative of L with
respect to V is the RP now just you
should probably remember with the indian
some velocity and in Hamiltonian circuit
Lagrangian mechanics keep in mind that
the derivative of the Lagrangian with
respect to a velocities of momentum this
is just
let's apply this to Lagrangian mechanics
in a few minutes but this is not a new
relationship for us now of course we
should use partial derivatives because
I'm imagining many variables parcel of L
respectively that's piece of our
definition at this point then we could
start H of P naught on a single momentum
but of a collection of peas
HNH of all the peas and this becomes
summation over all of the eyes piece of
ID sub I minus L this may start to look
familiar to you
let's go through the same little
derivation
they're the change in H is a summer
things a piece of ID so why I'm not
going to write summation signs whatever
I write a piece of I and V sub I I like
that and so forth think some all over
the islands alright so first of all a
piece of I dub the peace of mind of V
sub I of the peace of mind and then
derivative of L with respect the V sub I
times Delta V sub IV when I change
things a little bit this is what happens
to age again exactly the same pattern
the L by V G sub body is the same as
piece of iron so they cancel things
looks like great cancel and all I'm left
with is that the change in H does equal
I be variable X the change from this you
can easily read off with respect the
Peace of RT is equal to V sub I in other
words the change in H if you only change
one of the peas let's say the third P
the change in H if you've changed piece
of three a little bit is just V sub 3
keeping all the other peas fix again
same kind of reciprocity the L by V V
sub I times P sub I the age bracket P
sub I is visa body completely
symmetrically related to each other now
at the moment these are just abstract
variables
sometimes I've called one of them our
momentum
one of the new velocity but they could
be just abstract variables
and as I said there are many
applications of a change of variables
from the V type variable to a P type
variable that's kept track oh by r are
usually called generating functions long
interest is in mechanics mechanics the
Lagrangian is not only a function of
velocities it's also a function of
position we haven't talked about
positions at all we have velocities here
the things I quote V and where things
are called the relationship between peas
and bees and lagrangians and
hamiltonians is exactly what everything
but with the coordinates cute just going
along for the ride just going along as
if they do is if you didn't care about
them as if they were just passive so let
me show you how the relationship between
hamiltonians and lagrangians is nothing
but they were genre transformation of
exactly this type with the extra headed
ingredient that the small variables that
go along around when we're doing
mechanics you have a Lagrangian the
Lagrangian is a function of what the
function of the Q dots but let's call
the Q dots B's velocity generalized
velocity V sub I you survived on is by
definition V Suvari so the Lagrangian is
a function of the V's and the cubes
don't confuse Q's with peas
[Applause]
Levi is equal to Q dot we will write up
here another equation of mechanics that
you're familiar with P sub I is the
derivative of the Lagrangian with
respect to Q dot remember that the L by
D Q dot defines P but that's the same as
saying the L by DV equals P sub I that's
the first of our equations over here so
we start to see the Legendre
transformation between peas and
velocities beginning to say problem and
it's simply that it's simply the
definition of canonical momentum as the
derivative of the Lagrangian with
respect to Cuba next let's define the
Hamiltonian and absolutely repeating
myself now but I'm repeating myself in
the language of mechanics let's define
the Hamiltonian H to be the son of piece
of I P sub I of what P sub I and V sub i
we usually we write a piece of I Q sub I
got right that's language that we've
done previously but Q sub icon is
nothing but V surviving so let's put an
e surviving - for budget that's
precisely the same definition that we
had for the Hamiltonian I think in the
last two letters so you see
but now let's calculate the small change
in H when we change things what things
were going to change the peas and the
Q's what happens if we make small
changes
P's and Q's how does the Hamiltonian
change let's calculate that you're
taking the velocity to be a function of
P thank you it is a function into
another version generally that's right
in general you can think of Q is just
going along as the rougher a ride as a
set of parameters right the velocity
depends Roman but yeah that what P the
whole thing so that's correct
now said was correct rephrase it but let
me just restate it the relationship
between the peas and the vi's is such
that P as a function not just a V but
also the Q's and V is a function of the
P's and Q's but you can simply think of
the Q's as being parameters at this
point other said go along for the ride
the point is of course when you
calculate the L by V V sub I in general
it might contain some Q dependence and
not just be dependence but that's okay
okay let's calculate the change in H
when you change things a little bit
change things need to change
everything's that P's and Q's change an
inch same things before is first of all
the sum which I won't write the
summation sorry P sub I Delta V sub I
plus V sub I Delta P sub I now we have
minus the derivative of the Lagrangian
we have two things that they're written
Lagrangian depends on the other by the Q
sub I Delta Q sub on and - the L by V
sub I Delta V survived Lagrangian is
depend on Q's and B's and now we're
burying everything excited
these are cubes you don't have to
separately vary the velocities because
you know the velocities if you know the
P's and Q's so there we are that's the
formula for changing H a little bit and
have several terms B sub are thought to
be survive you survive over antibody
derivative will around you move through
step cube times of changing Q the rid of
the Lagrangian with respect to V don't
go there okay it's a little bit
complicated before turning three big
pieces of it just cancel the same place
over here remember that the L might be V
sub R T is P sub R is P sub I is P sub I
things in the red squares
the other two terms can't cancel each
other one involves Delta P the other
involves Delta Q those are independent
variations that can't cancel each other
oh we can write that now is the
derivative of the Hamiltonian with
respect to P sub I that's a small change
in H when you change P sub I a little
bit keeping Q sub I fixed the H breaking
P sub I to keeping the Q's fixed just
one what do you read off this V sub I
and the H might be Q sub I was that
this is minus PL by DQ survived that's
no problem okay that's not bad
nice equations a little bit okay over
here is equal to what's the we have to
elevate the Q right that's just equal
minus P dot that's the ranchers
equations
PETA is the old lady Q so we have one
equation let's get rid of the middleman
here we don't need them the H by D Q is
minus the time derivative of
oh it's beyond you doc
let me go through this again I'm gonna
do anything for these of course our
Hamilton's equations they're quite
beautiful H is a function of easy Q's
that's the way you think about it the
Hamiltonian is a function of these
accuse momenta and coordinates you get
rid of the velocities replace it by P DS
and then the derivative of the
Hamiltonian with respect to P's defines
the Q x and the derivative the
Hamiltonian with respect to Q's is minus
the peanuts that's the structure of
mechanics in the Hamiltonian nutshell if
you like let's go through it again just
to watch a series of steps all of which
are abstract none of which are highly
intuitive let's go through it again here
we go reach is equal summation piece of
IV survived - l.l.bean a function of Q
and V now let's make a little variation
that's going to be equal be some
identity sub body let's B sub R Delta P
sub I minus the change in the Lagrangian
what's that sorry but now we have to
calculate the change in okay first of
all we're capsules the biggest
multiplying not to be canceled why do
you have peace of already here
the Omega V sub I that's piece of our
definition a piece of art that's always
left over and we see the derivative of H
with respect to P sub I is V sub I but V
sub I is Q dot then we look at the other
one that says the derivative of H with
respect to Q sub I was equal to minus V
Omega Q Suvari - but then we use the
lungis equations the elephant DQ sub I
is D by DT of P sub I that's the ranch's
equation or just minus P dot
so we see that mechanics in great
generality gets repackaged as a system
of first-order equations very great
simplicity we need to know one function
of P's and Q's knowing that function of
P's and Q's immediately by
differentiation tells you how the point
moves through phase space it tells you
the velocity through phase space you
Dodd of course is just the ordinary
velocity but the combination Q dot and P
dot let's put the assign on the other
side over here the combination Q dot and
P dot is a kind of velocity through the
face face it tells you if you know P and
Q you plug it in the H calculate the
HDPE hdq if you know P and Q then you
immediately know P dot Q dot
reduces mechanics the system of first
order equations for P's and Q's every P
has a partner called Q and riku has a
partner called P they come come in pairs
and the hamilton equations are one sorry
two for each are a pair of equations for
each arm let's do it just for simplicity
for practicals they're made let's just
check that it really is correct
as I said I have no idea what a mobile
motivated Pendleton to write down but
they certainly do have a very very
simple structure now one of the things
that show you is that surprisingly with
very very different physical
interpretation P's and Q's seem to be
very very different things yet there's
this enormous symmetry between P's and
Q's they see the current in the
equations in an extremely parallel and
symmetric way apart from a single minus
sign over here they seem to enter in in
a symmetric way into mechanics now it's
the study of these equations which
really defines modern mechanics
let's draw a picture to indicate what's
going on
yes general face face paint killed it's
not just one variable as many variables
some subset of herpes every second for
the queues or can't throw them all on
the blackboard so this one people in
queue at every point and every point I
could calculate the HPP in the H by PQ
and tell you exactly what cubed on earth
keypad are that defines our kind of
velocity vector through the through the
phase space not an ordinary velocity but
a phase space velocity let's draw that
face phase velocity is an arrow it's
showing which way that point is moving
in the phase space and the length of the
arrow we can take to represent how fast
is moving through the phase space so
what the Hamiltonian gives you is a kind
of flow through space
the hours long going fast or the hours
are small you move slowly and the
velocity as I said is related to
derivatives notice one funny thing that
the Q velocity is a P derivative and the
P velocity is minus of Q derivative so
it's not that Q dot is the derivative of
H with respect to Q and P thousands of
variations expected beep to get the
frost yet they get crossed in the
equations that has a profound
implication as we'll see in a moment any
questions about Hamilton's equations any
second-order system can always be just
by defining the first derivatives to be
new variables then that simply defines a
set of twice as many that's trivial
but this elegant structure here is far
more than that it has some properties
that we'll see it alone when it happened
is that usually badly defined systems
for which the evolution is not
well-defined and usually reject it as
being the I mean are there any systems
in nature which require
well first of all friction is outside
the boundaries of the Ranji and the
Hamiltonian mechanics now not really
because all systems in nature are of
this type and friction is a part of
nature but as emphasized over and over
again you don't really see friction in a
system unless you ignore a large large
number of coordinates friction is simply
the excitation and the moving around
that microscopic coordinates you know
Franco so yeah the let me see I'm trying
to think are there any systems that I
can think of well you can't
invert the relation between coordinates
and yeah I can think of warm no
I can't think of any offhand that make
physical centers we'll talk about why
and the importance of this no that's
fiction
that's picture is happening
you cannot write equations with friction
in this form friction outside in the
sense that erupt the equations that you
normally write down to the friction they
won't be of this form if you write the
equations governing all of the molecules
every single one of them keep track of
them all then there's simply a huge
number of equations which do have this
form so introducing friction is a way of
approximating a very very large number
of equations of this form by a very
small number of equations and ignoring
the motion of most of the degrees of
freedom energy is leaking off into other
degrees of freedom collection we don't
want to be able to now picture is
exceedingly complicated
that's correct
okay well yeah we did the old Mikey B is
P and the H by D here this is Lewis on
the transformation it's taking the
father it's taking the function of V
it's a procedure it's a procedure of
changing variables if you like changing
variables from a set called V to a
second the transformation is a whole
bunch of arithmetic it goes into it you
have a pair of variables V and P they're
connected to each other and they're
connected in the one-to-one way then you
can define to find an L of E and an H of
P L of E is the integral under this
curve out to some beef H of P is the
integral over this curve up to P the
derivative of L with respect to speed is
P and the derivative of H with respect
to P let's be
e times V is Ella finish that's the
simple geometry of it why is it so
useful you get used to it then you will
you do it over and over again and you
begin to see why it's your school I'm
going to suggest we just go with it
and we will find out just tell you
school questions are which one you have
a bad system physics
[Music]
well usually means the equations
emotional breakdown at some point some
break down and predictive 'ti of the
equations singularity happens we can we
can write down some lagrangians and
hamiltonians which are bad and try to
see why they're bad I think that's
that's a good thing to do I'm not gonna
do it tonight some somebody send me an
email just send me an email saying write
down some bad systems and we'll write
down some examples when things don't
work out well and try to follow them and
see where the motion breaks down see
what happens to the motion power tells
you all of a sudden you don't know which
way to go
it usually means at some point the
equation of motion simply is singular
there's some infinity in it and
something breaks down because it
something in the equations comes into it
as a general rule it means that you're
all right now at least learning
Strasbourg means something can travel
with infinite velocity all kinds of
things can go wrong but I think it's a
good idea to take an example or two
which we you cannot invert the
relationship between velocity and
momentum and see it's not well first of
all we can have an infinite number of
degrees of freedom in the classical
mechanics infinite doesn't have to do
with the difference between are we
frightened of it it can get pretty
frightening having a field can meet the
friction for example a electron
accelerating radiates and radiates
photons the radiation of those photons
tend to act like kind of friction and it
can get very very complicated all of the
motion of an electron including
radiation reactions and so forth why are
we afraid we're not afraid of many
degrees of freedom we just have to
remember
I think you're asking me why it's so
terribly difficult to derive the laws of
friction and the laws of mechanics
well that is hard but also hard to
derive the laws of radiation reaction it
is hard any case but tonight we want to
discuss energy conservation among other
things we already discussed energy
conservation several partners but it's
interesting to see how these equations
form all from the front perspective
these now become the equations of
mechanics this is a new formulation of
mechanics based on H of P and Q given an
H of Q base space you have a motion
through the phase space you can imagine
dots in the phase space a fluid of point
in phase space moving each point with a
velocity that depends on its position
according to these equations and so it's
a flow of a fluid that's simply a new
way of formulating mechanics must forget
about all the old ways oh I said I
worked it out for you for for the
ordinary particle let's do that first
ordinary particle will get one-half MV
squared minus u of X what happened in X
dot squared now I've replaced Q by ax
check these equations so what's the
holocrons we've done that before P
squared over 2m plus you're next
B squared over 200 plus plus U of X is
the Hamiltonian let's just check
Hamilton's equations the H by BP
differentiate that that's P over here
that's where it is this peel of an
ending equal to Q got or in other words
is it equal to X stop yes it is our next
one is equal to P so the first of
Hamilton's equation is the H by DP is
equal to Q dot that's nothing but the
definition of momentum in terms of
velocity the work out the other equation
the H by the X minus the H by D X should
equal P dot the H by the X that's just
the U by DX u by D X well that's just
Newton's equations mass times
acceleration is equal to force so the
second equation typically the way it
works is the first equation is the
relationship between positions between
sorry velocity and momentum
it's just P over N equals x toposurface
here it's the second equation which
really looks more like F equals MA this
is mass times acceleration and this is
force okay so it does work it is correct
it is true then it is equivalent to all
the other formulations we could work out
some more complicated examples
well refresh let's concentrate on in
this new form let's concentrate on
energy conservation let's see if we can
prove directly that energy is conserved
let's calculate the HDP find derivative
of family if the Hamiltonian is energy
which it is then it should be conserved
its time derivative should be zero let's
calculate it is equal to the derivative
of H with respect to P times P dot plus
the derivative of H with respect to Q
times Q dot
[Applause]
sometime a heart change an age when you
change P times P dot plus the change in
a treaty change Q times $2 that's the
change in our funding per unit time but
now let's use Hamilton's equations for P
dot Q dot what is peeta peeta the age by
D piece of art that's this but what is
piece of our table can somebody read it
to me it's just a sign that I'm gonna
bat - that's - so - the H by D Q right
and how about this one this is plus the
H by DQ H by D P sub I and they cancel
that's what the minus sign is doing that
I think it's possible to imagine
equations which have a conservation of
energy which are not at the Hamilton
form if you have Hamilton's equations
with a time independent Hamiltonian I've
assumed the time I pursued was no
explicit dependence then you do get
energy conservation the other way no I
think you can know you can certainly
have equations which have conservation
laws conservation of an energy without
them being of Hamilton's form so it's
not
all right so we see that these two terms
simply cancel each other and energy
conservation is a very very simple
consequence what this says is if you
take a point and you follow it now no
one I want you to think about the space
space there's a collection of points
that you can follow in time as you
follow the entire you stay on a contour
of Christ and energy imagine the space
space and imagine it being filled with
contours of energy in other words
surfaces on which the energy is constant
simply build up the theis point meaning
a point in this base space as you follow
it stays on the same surface of constant
energy that's just energy conservation
and Luud flows the fluid defined by the
phase space thus think of it thus if a
space flows along these lines which are
lines of constant energy the lines of
constant energy in general could be
closed they don't have to look like this
it could be closed curves for example we
study the case last night called the
harmonic oscillator in which H was
proportional to P squared plus x squared
well let's forget the constant
forgetting the constants it was
essentially P squared plus x squared so
in that case the contours of constant
energy which is circles around
the contours can be open or off to
infinity they can be closed curves they
can even be about the corresponds to a
motion which doesn't go anywhere is it
just sits there for example the harmonic
oscillator which has neither position
which is at the origin with no velocity
and simply sits there it doesn't move
that's a very generous circle so the
phase point moves on these services this
is very much like the discrete situation
where you have different closed pieces
of the phase space and that provides a
conservation law which one of the cycles
you are closed Cyprus
well if a space is very similar every
point is on some kind of constant energy
and you move along those contours yeah
we haven't discussed effective defensive
points constant let's let me answer your
question
let's imagine why is it look like this
so the fluid comes in and goes through a
narrow of channels what happens when
fluid goes through a narrow channel
moves faster it moves faster just so
that the density doesn't change a big
lump of fluid comes in and it doesn't
want to change its volume so it has to
squeeze between these lines faster
unless it is where the where the lines
clump up and get close where the
contours get closed the fluid is moving
fast that's in fact these equations say
when the contours are close together
think about a contour map what does it
mean in a contour map that the contours
are very close and steep
it means the altitude is changing very
fast what does it mean on the contours
of constant energy that the contours are
very close together it means that H is
changing rapidly with respect to either
P or Q or some combination of P and Q so
where the lines
combo is where the H by P P and the H by
D Q is large and it's where the fluid is
moving fast so once we diverge alright
it's just that the places where they
diverge like that the fluid slows down
we're going to study that behavior a
little bit
we have closed curses what's that mean
you have closed curves to ugly let me
give you an example
the pendular pendulum
small oscillations at the bottom of
closed curves the system just goes back
and forth back and forth two cycles over
the same swings over the top like this
they sort of correspond to open curves
well take a particle that's moving on a
washboard potential like that potential
which is just a sine wave if you stop
down near the bottom yeah and you give
the particle a little bit of energy
all it does is oscillate back and forth
so it looks like a harmonic oscillator
in fact it is a harmonic oscillator plus
a little bit of extra just oscillates
back and forth and the contours are
closed curves so if a small energy
contour is a closed curve give it a peak
energy and it just goes whistling off
over the top of the washboard if it has
enough energy to climb over the top and
keep going
then the curves are open the system
doesn't come back to itself it just
keeps going off forever and ever so yes
you can certainly have both closed and
open curves in the same phase space
we interested in to China I'm not going
to do it they'll be interesting to try
the plot what the contours of constant
energy look like a space any other
questions
or anything you want me to go back over
yes
it's just we can invent all sorts of
panettone H is equal instead of P
squared plus Q squared let's take a cube
X cubed u 7p squared plus x squared this
doesn't correspond to any system that I
know about a nature but this nothing
wrong with it from the point of view of
the Hamiltonian one of the equations the
equations are the derivative of H with
respect to P is equal to X bar and
that's three P squared is equal to X
stock well that does not say that the
relation between P and X table is that T
is a constant multiple of X time so
that's not nothing general other this is
sort of unusual but you couldn't have an
X here that would be less unusual no
sorry x times P squared also x times P
is occurs with kind of programs next
time then you are have the H by D P is
is x times P and you would have P and X
the relationship between them having an
extra factor of X so now
there's no reason for such a simple
relationship motion is perpetual motion
this perpetual
that's why energy is conserved but
before we can address the real meaning
of perpetual motion and the pros and
cons and so forth we have to understand
something about thermal dynamics and we
have to understand the difference
between usable and non usable energy and
entropy and all that sort of stuff so
we're now at a more basic level studying
the individual degrees of freedom of a
system microscope for example if it's
table with all the molecules moving
around and so forth of the air in the
room these are the equations that really
do government the entire sister in there
is kinds of approximations you divide
energy into usable energy or available
energy and unavailable energy heat is
unavailable energy but that's got to do
with the second law of thermodynamics
which we haven't got to yet I would hope
to get - it would be nice to be able to
get to it it's deeply connected with
these equations though it's deeply
connected with the structure of motions
and phase space
before we understand that thermodynamics
we have to understand the chaos we are
not ready to do that tonight hopefully
we'll get through it
okay so let's discuss other conservation
laws energy is a conservation law that
followers just from Hamilton's equation
see if there are other conservation laws
but let's look at the general form of
you know from the Lagrangian point of
view the conservation laws have to do
with symmetries I'm going to suppress
that now
later on we'll come back to discuss
symmetries from the Hamiltonian point of
view but now let's not worried about why
there might be a conservation law just
let's see what it means to have a
conservation law what the necessary and
sufficient conditions for conservation
is conserved what does a a is a function
of the phase space wherever you are in
the phase space a has some value it's a
function of the peas and the Q's or
equivalently a function of position and
velocity let me give you an example we
worked out number of times ago that the
angular momentum of a particle moving in
a plane is given by x times P y minus y
times PX it's a function of the peas and
the curious general all quantities that
we might be interested in or functions
of positions in momenta if we know the
position and everything of something you
know everything about it
so in particular we know the value of
its various properties let's ask for the
eight of P Q might be conserved might it
be concerned under what conditions
mathematical conditions now I want to
express those conditions in terms of
Hamilton's equations that's going to
introduce for us a new set of quantities
while plus some brackets
fish brackets
okay this is easy what we want to know
is whether the tiny derivative of a is
zero in fact it's just calculate the
time derivative of a be a by DT let's
just do a crude forceful calculator it's
the derivative of a with respect to P
sub I times P sub I dot plus the
derivative of a with respect to Q sub I
times Q so Phi dot I'm gonna write the
summation sign explicitly
but now what substitute in the piece of
I thought inclusive I've got their value
in terms of the Hamiltonian so this is
our derivative of a with respect the
piece of already
what's piece of I got again a sign I can
never remember the design it's - right -
the H by D Q sub I so we have the 8090 P
sub I the H by D Q sub I now what's this
one
plus the a by Q sub I cubed R that's the
HDP right
well okay that's kind of interesting we
find out it's given by the sum of two
terms let me start with this one the a
by D cube the H by B P so if you know
what a is and you also know what the
Hamiltonian is you can compute these
there's the a by DQ D H by D P minus da
by D P D H by D few a very symmetric the
put them inside a bracket and put a
comma between them and that's the that's
the super it's definition definition is
minus
maybe by the key survive
you can see the symmetry of it it's
composed of two terms one in which eight
gets differentiated with Q when with a
beat gets differentiated with P and vice
versa that's called the Poisson bracket
of classical mechanics in terms of
whether it's that's another formulation
of mechanics is a special object which
generates velocities generates some
brackets
well they say this is another yet
another formulation for mechanics where
the motion of any quantity is expressed
in terms of let's just check it let's do
one or two more examples
if you like this is a generalization of
Hamilton's equations include tentative
places for example let's check you got
killed on must be the Poisson bracket of
Q with inch according to those equations
let's see what that says that says one
variable at one key in one cue that says
the Q by Q the age fighting PD - the Q
by V P the H by D cube I've now just
substituted for a to throw q q dot is
the Poisson bracket of Q with age the
Poisson bracket a few of h QD q what's
that one
what's the Q by DP 0 Q and V are
independent variables this goes away and
this just says the Q dot is the HP ATP
no that's Hamilton's equations what
about P dot P dot is the Poisson bracket
of P with age that's P by DQ
- BP by BP the HIV - what's the people
you killed you're all the people at this
peak just erased my inside so that's the
other of Hamilton's equations so the to
Hamilton equations are special cases of
a much more general rule trying to rid
of anything is the Poisson bracket or be
anything to the Hamilton excuse a new
view of Hamiltonians as generating time
dependence by the action of Poisson
brackets the action of taking the
Poisson bracket the Hamiltonian tells
you how systems change with time that's
yet another representation of classical
mechanics I think that's probably let's
so let's stop the question is enough to
join oh well since I started talking
about conservation laws and now we see
there's a necessary and sufficient
condition
a conservation law that the quantity in
question have zero Poisson bracket with
the Hamiltonian now let me see my
incredibly abstract view of things and
it is but it's also a very powerful and
very general population concern zero
then whatever it is for example
I really don't know I wouldn't be
surprised if you came up with it exactly
this way by calculating small changes
and things exactly the structure
recurring mathematicians good
mathematicians theoretical physicist
other people tend to see patterns they
see the same thing a couple of times and
there's a pattern I see a pattern what
are some other examples were
oh wait till you're talking about non
conservation of energy thread y'all good
okay I have a theory about the
Lagrangian and do not explicitly depend
on time now by explicitly depend on time
it means that the some time dependence
above and beyond the fact that the Q's
and the fees or accusing the cubox
depend on fire let's go and ask what
happens
if the hamiltonian really does depend on
time how can that happen well what
suppose is a big welcome matter over
here our system is over here we're not
going to include this as part of the
mathematics of our system where the
exert forces on now will be the forces
on the system will depend on time in
fact there will be eternity energy which
is a single particle but the eternity
energy of this particle you'll of its
position which also depends on the
position of this big lump but the
position of the big lump depends on time
so that means the potential energy
depends on X and also explicitly primary
it means it depends of time even if the
particle doesn't move it certainly
depends on Pine just because X may
depend on time but that's an implicit
dependence on time there's also an
explicit dependence on time
corresponding to the fact that the
facility of the force depends explicitly
on time let's go back
[Applause]
let's calculate entire derivative of the
energy itself
this was our equation that we had a few
minutes ago and these two things cancels
because P dot Q dot r also derivatives
of age but that's just another term if H
also depends on T then we have to write
that the prime derivative of H is the H
by V P times P back plus the H by D Q
times Q Bar plus they explicitly
derivative of H with respect to T in
other words the change in the
Hamiltonian
even if P and Q were standing still P
comma Q the others are equal to 0 that's
what happens if the potential energy for
example depends explicitly on time then
this would of course be just the u by DT
so in that case even though Eastern
terms completely cancel each other by
virtue of Hamilton's equations there's
another term left over just the explicit
derivative of H with respect to fear and
that's in general non zero there won't
be zero familar cookie in separate sense
if the Hamiltonian explicitly depends on
party and that will appear in the
equation for the time derivative of
total energy so that's as simple as that
because we can do some examples it's
getting a little bit late
I was just
[Music]
we could define it by da by D PUD
barbecue - the other one but this is the
way it's defined it's not like that's
not like question is in city centers you
have a function are you taking the
derivative of a with respect to the
first variable or to the second variable
in your in your capital letter
this is P squared over 2m plus 1 of X
just tell you where the X is the first
day you know I don't want to
differentiate something
momentum in position that there's only
one way to say it's not my question
Hill you said as an abstract mathematics
ed he said that this Poisson brackets
gonna come up in all kinds of places you
know maybe that's not momentum in
position in which case it doesn't matter
which is which variable this particular
structure doesn't come up anywhere
except about mechanics look some people
seem to be under the impression that
it's important to describe where the X
is the first variable on the second
variable it's not it's just the X
variable and B is the P variable when I
calculate the H by T P I don't have to
ask where the P comes first into X
second that's all for now
you could have defined the Poisson
bracket in the opposite lane in fact it
could have defined the Hamiltonian of
the opposite sign what kinds of things
you could have done differently we made
some convention this is one of the
conventions
all over the place and I didn't know
that is always yeah I don't know any
other place but some records come up all
right
except for positions and momenta I did
say comes up over all of those places
with Chandra transformations which are
the transformations do have this habit
of coming up all over this place the
questions to Lassonde transformations
don't have a funny - I remember I read
the L by V is equal to P the H by V V is
P is equal to B so in that particular
case there is no
[Music]
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