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classical mechanics is the basis for all
of physics it's the basis of all of
physics not only because it describes
the motion of objects like particles and
mechanical systems and so forth but
because the basic framework the basic
structure of all the physics is based on
the principles of classical mechanics
the conservation of energy the
conservation of momentum the principles
by which all systems evolve in nature is
the same set of rules essentially
exactly the same set of rules in a more
abstract in the more general setting
then the rules which govern how a simple
particle moves for example under the
influence of gravity but in order to
understand that we have to understand
the principles in a fairly general
context let's begin with the very very
simplest kinds of systems that we can
think of systems that are so simple that
in fact is simple than any real systems
in nature laws of nature let's imagine
laws of nature which are of the most
primitive and simple kind acting on the
most primitive and simple systems that
we can imagine I want you to first of
all suppose that time which evolves
continuously under the my watch I see
the second hand goes around and around
and around that goes around continuously
the time can evolve and be any real
number t I want you to imagine it only
occurs in beats a stroboscopic world in
which you only look at the world at
intervals of time which we could to be a
thousandth of a second or it could be a
millionth of a second let's just take it
to be one second intervals and ask how
in the stroboscopic world systems change
with time let's also imagine a very very
simple set of systems systems which are
so simple that they only have a handful
of configurations configurations mean
everything we need
know about the system to characterize it
completely so the simplest system I can
think of would be a system that has only
two configurations heads or tails a coin
lay according down on the table I don't
have a coin so we'll take this coffee
top here I could put it down heads or I
could put it down tails at that if you
can tell the difference from where
you're sitting but I can tell the
difference this is this is heads that's
tails all right so we have a system then
it's characterized by two states two
states of being and we want to add to
those two states a law a law of
evolution in going from one instant of
time to the next instant of time from
one beat of the stroboscopic light to
the next one what kind of laws are
allowable what kind of laws the basic
principles of physics allow and what
kind of laws don't they allow that's
going to be our first kind of question
for tonight
so our first concept is the space of
states in this case just heads in tails
it's just two points heads and tails so
two points in an abstract space heads
and tails it's called the phase space
it's called the phase space of a system
the space of possible states of a system
and what do I mean by a state of a
system I mean everything that you need
to know in order to predict what happens
next
everything you need to know in order to
be able to say with certainty what the
next state of the system will be that's
called the phase space of the system in
this case just heads in tails what kind
of laws can you imagine what kind of
laws of nature can you imagine for this
extremely simple world it is the
simplest well I suppose you could
imagine a slightly simpler world you can
imagine a world with only one state
heads not much can happen in that world
there's only one law of physics heads
goes to head since heads goes to heads
goes to heads it's extremely boring
nothing ever happens because there's
only one state so how could anything
happen but with two states has entails
you can have a richer variety of
possible laws of physics one law let's
take the various varieties of laws of
physics we can have one law might just
say you stay the same if the state of
the system is heads at one instant of
time the next instant of time it will be
heads
if it starts tails it will stay tails
that's a very boring law and let's graph
that law by drawing an arrow
if it starts heads then it stays heads
let's just represent that by drawing an
arrow from head to head and from tail to
tail the meaning of this arrow you start
at the tail end of the arrow and you
follow it and you see that it comes
around to the same point that stands for
the rule that says that a heads stays a
heads and in this case a tail stays a
tail so
a dull law of nature this stays that way
for endless amounts of time and this
stays that way for endless amounts of
time another possible law of nature
would be slightly more interesting
situation if you look at it at one
instant of time the Nexen since it's the
opposite
heads goes to tails because the heads
goes to tails
these are deterministic laws
deterministic laws mean that if you know
what is happening at one instant you
know forever after you know everything
about the system infinitely into the
future completely deterministic
classical mechanics has that nature to
it that is completely deterministic so
and in fact it is in a certain sense
always reversible but we're coming for
that all right so to draw the graph
representing heads goes to tails and
Tails goes to heads we draw an arrow
from heads to tails and from tails -
heads and we read that as saying that if
you start at heads you go to tail if you
start at pals you go to heads
what is the evolution of a system under
this law of nature if you start with
heads it's heads tails heads tails heads
tails heads tails forever after these
are two consistent laws of physics in a
world of only two states whether we can
think of more laws yes we can but but
for the moment these are two interesting
ones now how can we generalize this we
can generalize this first of all to
systems with more States to States is
not very many we could have a die die as
in dice die has six states one two three
four five and six and we could represent
them as points six points now we have a
large variety of different laws of
physics that we could have for example
we could simply have we could label
these one two three four five and six
which one is one and which one is six is
not very important but the
there are six of them we could have a
law of physics which says 1 goes to 2 2
goes to 3 3 goes to 4 4 goes to 5 5 goes
to 6 and 6 goes back to 1 this would be
a complicated motion when thought of in
terms of a dye you start with a 1 up and
then it goes to the 2 up 3 up and so
forth but it's a relatively simple
picture when drawn in this manner here
you can imagine more complicated laws of
physics with six states well whether
they're more complicated or not is in
the eye of the beholder but there first
of all we could have a similar law where
instead of one going to two one could go
to three three could go to to two could
go to five one two three something like
that that's really not very different
than this each one goes to a neighboring
one well not on neighboring one each one
goes to a next one and they cycle around
and one characteristic of such systems
is they will just cycle around forever
and ever and ever
okay now you can have more complicated I
don't know if they're more complicated
different laws for example you could
have a law which goes this way if you
start at one you go to two if you go to
two then you go to three and then you go
back to one if you happen to start at
three note this is three this is five
right five six six six six one two three
four five if you start at three you go
to four if you start at four you go to
five back to three in this case
well that's somewhat similar to this you
have two disconnected cycles to do got
more complicated things here's another
one in fact I think it would take a long
time to draw all the possibilities one
goes back to 1 2 goes to 3 3 goes to 4
all goes back to 2
five goes to six and six goes back to
five right they're all acceptable laws
of physics wherever you happen to be you
know exactly where to go next
so it's deterministic it's deterministic
into the future meaning to say wherever
you start you know where you will be
arbitrarily into the future if you start
here and you go a hundred thousand times
you'll just wind up writing up some
wherever wherever you start here you
stay there so these are completely
deterministic into the future but
they're also completely deterministic
into the past which means if you know
where you are you know where you were
before if you find yourself over here
then you know in the previous cycle or
the previous instant you were over here
and so forth so you can trace your
motion either into the future or into
the past with complete confidence about
where you'll be no matter how far you go
that's the character in principle in
principle if not in practice and
practice things get jumbled up and you
can't see them clearly enough and you
and you miss detail but if you can see
the infinitely small detail in a system
and get every single bit of physics
absolutely right for any classical
system they are in this sense exactly
deterministic both into the past and
into the future now what kind of laws of
physics do we not allow the kind of laws
of physics that we don't allow I can
best illustrate I think by drawing some
some possibilities and explaining to you
why they are not allowed here's a law of
physics that is not allowed by the
principles of classical physics three
states three states is perfectly alright
nothing wrong with that but let's draw
some arrows - 3 2 goes to 3 and 3 goes
back to 2 that would be perfectly
alright by itself what about 1 well
I could have one goes to one but I don't
want one to go to one I want one to go
to two that's completely deterministic
into the future if I start at one I go
to two I go to two I go to three I go
three back two to two back to three I
always know where to go in the next step
I just follow the arrow wherever it
happens to be but what does it fail it
fails to be deterministic into the past
supposing I know that I'm at - then
where did I come from
I could have come from one or I could
have come from three so I cannot work my
way backward with uniqueness I can work
my way forward
I cannot work my way backward that's a
this is a law of physics which is
irreversible it would not allow me to
run the laws of physics backward it
would lead to an ambiguity every time I
were at to another law which is not
allowed by the principles of classical
mechanics or principles of classical
physics would be basically the same
thing but with the arrows turned in the
opposite direction all arrows reversed
here I have a problem not going into the
past but I have a problem going into the
future
let's say yeah supposing now I find
myself a - and I want to go into the
future I don't know whether to go into
the future by following this arrow to
one or this arrow to three there are two
arrows leading out of two over here one
of them goes to three one of them goes
to one nothing tells me which I wrote to
follow so it's not deterministic into
the future I might randomly decide to go
from two to three or randomly decide to
go from two to one these are rules each
one of which is deterministic in one
direction but not the other
these are the sorts of things which are
forbidden by the principles of classical
mechanics
why are they forbidden by the
principal's yeah which one yeah that
wouldn't come I wouldn't know where to
go from three you get stuck at three
yeah no good well you can't have a
dangling one you gotta know where to go
next
now you could let's see we could try to
put something like that in yeah okay so
I would say we go from 1 to 2 from 2 to
3 2 3 2 3 2 3 now the problem with being
going backward I think yeah yeah yeah
right so one way or the other you get
stuck over these rules all right how do
we spot what's allowable and what's not
allowable well it's very simple
at every configuration 2 or 3 we should
have one in and one out
we should have a unique one in a unique
arrow in and a unique arrow out one
arrow in to tell us where we came from
and one arrow out to tell us where we're
going
that's the character of classical
physics uniqueness into the future
uniqueness into the past when
represented in terms of this very simple
and analog world analog digital world of
our finite number of states then the
rules of physics as we know them would
say every configuration has one in arrow
in one out arrow now with this rule of
course life is very boring because if
you only have a finite number of states
all it can happen to you is you cycle
around endlessly among those number of
States always in the same order you can
have slightly more interesting
situations there's no reason why the
number of States has to be finite you
could have a situation where there's an
infinite number of possible state
a state corresponding to every integer
positive and negative a particle on a
position on a line where the position
could be any integer value and then a
simple law of physics would be you go
from one to the next wherever you are
you go to the next one each point has
one n in one out I can't draw them
endlessly it'll take forever but each
mark one in and one out this of course
would also be a rather boring law of
nature you just hop from one to the next
to the next and next and next forever
and ever and ever but at least you
wouldn't be cycling around endlessly
again we can have add some more states
on to this we could add this on if we
start on the line we simply move off and
keep going forever and ever and ever but
if we start over here we cycle around so
we could have mixtures of both kinds of
things some states we cycle around in
other states we move off to infinity now
notice that some of these laws of nature
the phase space breaks up into different
pieces which are connected among
themselves but not connected to each
other for example even with just two
states we had two possibilities one like
this and one like this in this case
where it breaks up into more than one
more than one piece we have something
called a conservation law a conservation
law is simply a memory of where we
started
a conservation law means that something
is kept intact for all time some piece
of knowledge is kept intact for all time
and doesn't change in this case we could
label this we could label a
configuration over here with a plus one
and a configuration over here with a
minus one
and then we could call we could invent a
variable plus one over here and minus
one over here it never changes if it's
plus one that stays plus one if it's
minus one it stays minus one that's a
conservation law something which doesn't
change with time on the other hand if we
hop from plus 1 to minus 1 to plus 1 to
minus 1 we don't have a conservation law
conservation laws are always associated
with these kind of closed families of
different trajectories in the phase
space which don't mix with each other
which remember arm something about the
system which might otherwise get mixed
up if everything got all mixed together
so there's all kinds of possibilities
that are inherent in these deterministic
laws but always the condition is one in
line and one out line for every point
that can also be called information
conservation its information
conservation in exactly the sense that
that you never lose memory of where you
started either into the past or into the
future if you know where you are at any
instant you know where you came from and
you know where you'll be information
about where you are is conserved never
changes into the past in the future
whereas if you have one of these bad
laws laws which are forbidden by the
rules of classical mechanics then you do
lose information for example if you find
yourself over here you don't know
whether you came from here or from well
no that's not quite right find yourself
over here you don't know whether you
came from here or over here so you lose
information information conservation is
perhaps the most fundamental law of
basic classical physics that you don't
lose information about now why that why
is that so it's not written into the
laws of physics why they are what they
are maybe someday we'll understand
Hyper laws
physics or metal laws of physics or
deeper laws of physics which will tell
us why the laws of physics are what they
are at the moment it's more or less an
experimental fact that all the known
laws of physics fit into this class of
information conserving laws even those
which are quantum mechanical but for our
subject this quarter we're only
interested in classical physics all
right so that's the basic set up if we
were interested only in this
stroboscopic world of discrete time
intervals the real world of course is
more continuous than that supposing okay
so let's make the law something like
this if the last two states were heads
heads then it stays heads if the last
two states were heads tails then it goes
to heads if the last two with heads
tails heads then it goes to tails and if
the last two with tails tails then it
goes to tails I think that's a possible
possible law then you would say I can't
tell from the fact that it's heads where
it goes next and indeed I can't but that
would just be another way of saying that
a specification of a heads by itself is
not what you would call a state where
you would call a state would be the
specification of the previous last two
entries because you need two entries to
tell you what happens next now that
raises the question that that's very
very important in classical mechanics
how much and what exactly do you need to
know to say what happens next
if the phase space is the space of
things space of possibilities but always
in such a way that they tell you exactly
what happens next what is it that you do
have to know next
so that brings us to continuous physics
let's take the motion of a particle
let's take the motion of a particle is
it enough to know where a particle is to
say what happens next let's hypothesize
that the generalization the continuous
time is that we need to know the exact
location of a particle along a line
we've run out of ink I'm afraid not
there is all right so let's imagine the
motion of a particle along a line then
you might think that the analog of a
state is just a location of a particle
where is it but is it enough to know
where a particle is in order to say what
happens next now what else do you need
to know its velocity in order to know
where a particle will be next you need
to know not only where it is but how
fast it's moving you need to know its
velocity so that means the state in the
same sense that I used it that which you
need to know in order to know what
happens next
does not just consist of the location of
a particle but you can say it two ways
you need to know not just the location
of the particle but you need to know
also the previous location or better yet
what causes for what is equivalent to
knowing the previous location the
velocity of velocity that tells you that
the phase space the space of states the
space of configurations is
two-dimensional not one-dimensional it's
not just a line it's a line that
represents the position of the particle
and a second line which represents its
velocity either to the left or to the
right positive velocity means moving to
the right negative velocity means moving
to the left supposing you're over here
where do you move next
you stay the same place why you're at
the origin and you have no velocity what
if you're over here so we could just we
could draw a little loop here to say
that you come back to the same place
what if you're over here you stay the
same place because you're moving with 0
velocity vertical axis is velocity so
you come back to the same place what if
you're over here where are you one
second later let's chop time up into
one-second intervals where are you next
somewhere to the right huh what if
you're over here now what if you're over
here move twice as far or roughly twice
as far to the right
what is it down here you move to the
left so we could fill up this space here
with little tiny arrows to show where
you move next but notice - no way you
move next you have to know not only
where you are in the sense of what X is
but you also have to know the vertical
component namely what the velocity is so
that means the analog the analog of a
point in the phase space is a point in
the space not only of positions but also
velocities yes why
where that take place it means no it
means exactly what the gentleman asked
me before what if you had a law of
nature which tells you in order to know
where to move next you have to know your
previous two entries all right now one
way of saying it is all right in that
case I need to know the previous two
entries and it just doesn't fall into
this class of things or you could say
that the space of configurations doesn't
just consist of a heads or a tails but
it consists of a pair of entries a pair
yeah that's right we meant we're not we
model that that's right we need two axes
that that's right we need an axis for
the present configuration in the past
one yeah that's exactly right well of
course not because in classical physics
the position of a particle and so forth
is a real number a real number is
something that you can never determine
exactly ah and so there's always
imprecision and that imprecision always
represents a degree of uncertainty and
where you will be next now it gets worse
and worse in general it's likely to get
worse and worse as you try to take
larger and larger time intervals a given
degree of imprecision in what you
actually know can get magnified and get
the magnified into worse and worse
imprecision as time goes on so are in
practice in practice classical systems
don't really have the property that you
can predict endlessly where they're
going to be and exactly what they're
going to do but you can always say given
a time interval I want to be able to
predict exactly for the next 30 seconds
where every molecule in this room will
be let's forget quantum mechanics now I
want to be able to predict exactly where
every mile
kyoool there will be then there's a
certain degree of precision in the
present information at exactly one
instant of time which will permit me to
be able to extrapolate for 30 seconds
okay it will not permit me to
extrapolate for 40 seconds if I try to
extrapolate for 40 seconds I will find
the errors get magnified out of control
if I want to predict correctly for 40
seconds I will have to do even better in
my initial conditions and my knowledge
of exactly what the state of the system
is so in practice this idea of
determinism is defective it's defective
because in order to determine for a
given length of time you'll have to have
precision which is so good that it's way
way beyond anything anybody can do
but in principle given any length of
time in classical physics there exist a
degree of precision which allows you to
extrapolate for that length of time does
that answer the question
yeah ah so some systems are very
predictable some systems are less
predictable and get out of control very
quickly they're called chaotic systems
but the principles are the same that
it's just a degree of precision which
you need to know in the beginning in
order to extrapolate for a given length
of time well you could decide I'm not
sure what the difference between
predictable and detail the way I use the
terms I use them interchangeably so I
can't say that
okay okay yeah all right what's that
yeah right the equations are
deterministic if you know the initial
conditions are deterministic are
predictable
infinitely predictable if you know the
initial conditions with infinite
precision you never do and therefore
they're never completely predictable now
the fact that you need to know both the
position and the velocity in classical
theory in the physics in order to
predict what happens next reflects
itself in the structure of the equations
of mechanics specifically it tells you
that the equations of motion Newton's
equations in this case are what are
called second-order equations instead of
first-order equations let me illustrate
it by starting with a first-order
equation a first-order equation what a
first-order equation means is that it
only has quantities of first derivatives
with respect to time in other words only
contains velocities second-order means
it contains not only first derivatives
but second derivatives means it contains
accelerations the equations of motion
acceleration is the second derivative of
the motion um we could write we know
what the real equations of motion of
Newton are F equals MA it contains
acceleration let's write a phony
equation let's write F equals mass times
velocity force is F velocity is velocity
and let's suppose that force just
depends on where you are we have a
particle that moves along a line it's
subject to four
which vary along the line force may be
big here small there and so forth so the
force depends on position and let's
imagine this fake equation of motion
that it's equal to mass times velocity
and what is velocity velocity is the
time derivative of the position the X by
DT and I will use continuously
throughout this course the notation that
time derivative is just indicated by a
dot that means time derivative all right
what does this tell me what do I need to
know in order to predict what happens
next I say for this equation we only
need to know the position of a particle
if we know the position of a particle I
can tell you what the velocity is just
from the equation if I know that the
position is a particular position then I
know the force on it
and from the equation that tells me what
the velocity is does it tell me the
acceleration how about the acceleration
let's see if I can compute the
acceleration to compute the acceleration
from this equation we just differentiate
it once more we write that the F by DT
is equal to mass mass is just a constant
times the second derivative of the
position or just the acceleration so we
have over here acceleration what about
the F DT the time derivative of the
force the force varies with time because
the position varies with time so the F
by DT is just a reflection of the fact
that the position of the particle varies
with time and we can write the F by DT
using standard rules of calculus as just
you did the change in F with respect to
position times a change in position with
respect to time in other words the
velocity alright we've already figured
out what the velocity is knowing the
force knowing the position so we know
the velocity
and we can read off from this equation
what the acceleration is
we can figure out all of the derivatives
of the motion if we know where the
particle is by multiply differentiating
this equation so what it tells us to
make a short story out of it is it tells
us that if we know where the position of
the particle at any instant of time then
we know where it's going to be in the
next instant the next two instance the
next three instants it completely
predicts the motion but this is not the
character of Newton's equations Newton's
equations say F is equal to mass times
acceleration mass times velocity so
let's look at this equation F equals
mass times acceleration can I predict
from this what the velocity is no this
is no equation for the velocity right if
I know what the position is I know what
the force is that tells me what the
acceleration is but there's nothing in
these equations which tell me what the
velocity is that means I have to add in
the velocity as a piece of information
to begin with I have no choice I have to
tell you in order to predict I have to
tell you the position as well as the
velocity then if I know the position and
the velocity I can then predict the
acceleration the next to a third
derivative the fourth derivative and all
of them so that tells me that in order
to know where I am and where I'm going
to be I have to know the position and
the velocity the phase space is a two
dimensional space so we see then that
that classical mechanics does have this
character of a configurate or a phase
space of different configurations with a
set of little arrows which tell you
where to go next
but the phase space itself has a
position component to it and a velocity
component to it we will go on and study
the classical equations of motion and
study them in a variety of different
formulations but always the
connecting link will always be
conservation of information the idea
that the laws of physics are completely
deterministic and described by equations
which tell you where you will be next
that's the character of classical
physics okay bifurcated it's one system
but you need two pieces of informations
that are warn you I'm not sure what you
mean by bifurcated up here here you only
need know now here you need one piece of
information to say where you'll be next
if you here you stay here if you're
there you stay there yeah the stuff all
there is is heads and tails that's all
there is and if you know where you are
you know where you'll be next
no no we're in oh no no no no this one
rule
know this this is one rule heads goes
two heads tails goes two tails that's a
single rule and if you know that you're
at heads then you know you'll be next at
heads if you know you're at tails
you know you but so you only need the
piece of information heads or tails that
does it that tells you everything now we
I suggested a different does well yeah
you don't need to know where you were
before you only need to know where you
are now to know where you'll be next so
both of these laws require only
knowledge of where you are at one
instant to tell you where you'll be next
you simply follow the if you start
someplace you follow the arrow until you
come back or you follow the arrow till
you get to the next place and that tells
you where you'll be next you don't need
any more information than that the only
question is what one of these points
corresponds to does one of these cut
points correspond to how much
information does it correspond to a
point in this space
is it enough to know heads or tails to
know what you'll do next or might you
need to know the two previous things
that's a different that's a different
set up weighted stay with the heads and
tails for a minute let's suppose that in
order to say what happens next you need
to know the previous two configurations
let's do let's work out that example and
let's write it in this form let's make
up a law I think I had one down before
heads heads goes - heads heads tails
goes to watch tails tails heads goes to
heads and tails tails goes to tails okay
is there something wrong with that oh
this only depends on Ed's coffee right
you're right you're right right sorry
sorry sorry sorry heads yeah we want
this one to go to tails uh second one
goes to heads good yeah what what what
what say it again it's also true yeah
yeah yeah yeah it's it also hard to work
an example isn't it let's see hmm
switch the second one this one make it
tails okay I think that's perfectly all
right
you lose information why well okay
here's a here's a set up here's a set up
if you know the previous two then you
know what happens next in fact let's
let's now see if we can make a table out
of this not a table but a set of points
the set of points then there are four
possibilities heads heads heads tails
tails heads and tails tails okay let's
see has heads heads heads tails tails
heads and tails tails now supposing you
go from heads heads to heads
this is heads heads heads that means you
go from heads heads to heads heads right
you go from heads heads to heads heads
now supposing you go from let's see what
happens if you start with heads tails
where do you go to so you go from heads
head tails - tails tails okay okay what
happens if you go from tails tails are
we going to run to trouble aren't we
tails tails goes to tails so tails tails
goes to tails tails bad not allowable by
the laws of physics and now tails heads
goes where tails tails goes to tails
tails great tails heads tails heads goes
to tickles - heads tails so that comes
here well it seems to me we go we should
be able to make a consistent law whoa
what uh this one this one
Oh tails tails goes to tails tails goes
to heads yeah
okay so then tails tails goes to heads
and that means that tails tails heads so
I'm his tails tails goes to Ted - tails
heads ah oh okay good good now we have a
workable law now we have a workable law
the only thing we had to do was to say
that what we originally called a
configuration namely a single heads or
tails was not a complete specification
information a complete specification of
information involved
two pieces of information and once we
recognized that we were able to to to
write this down as a law of physics
which is deterministic and reversible
okay so you don't know offhand you don't
know to begin with what pieces of
information you need to know in order to
know what to do next but that's what a
stead that's what the configuration
space or that's what the phase space is
it's the collection of all the things
you need to know to know what happens
next
now of course you could go beyond this
you could say I need to know the first
three things in order to know what
happens next you can do that you'll
simply need more states to make a
deterministic system what would it mean
in classical mechanics to need more
information than the positions and
velocities suppose you need the
positions velocities and accelerations
that were in third order differential
equations but what would it say about
the phase space it would mean you would
need positions velocities and
accelerations to represent the phase
space as it happens that is not the case
for classical mechanics that's an
experimental fact but it wouldn't stop
us if we did if we did need the
accelerations we would just write third
order equations and we will make our
phase space three-dimensional
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