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classical mechanics is the
basis for all of Physics this basis a ball physics
not only because so it describes the motion
objects like particles and mechanical systems and so forth
but because the basic framework the basic come
structure all a physics is based
on the principles of classical mechanics the conservation energy the conservation
of momentum
are be principles by which
all systems evolving nature is
the same set rules essentially exactly the same set of rules in a more abstract
in the more general setting
in the rules which govern how simple particle moves for example under the
influence of gravity
but not to understand that we have to understand the principles
in a fairly general context are let's begin
with very very simple worst kinds of systems that we can think of
systems are so simple then in fact this simple than any real low our systems in
nature
laws of nature to imagine laws of nature which are the most primitive
and simple kind acting on the most primitive and simple systems that we can
imagine
I want to the first the wall suppose that time
which evolves continuously on the my watch to see the second inning goes
around around around goes around continuously
the time can evolve and be any real number:
T I want you to imagine and only occurs in beats
a stroboscopic world in which you only look at the world
at intervals of time which we could be a
thousand for the 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
out very very simple set of systems systems which is so simple
but they only have a handful of configurations
configurations mean everything we need to know about the system to characterize
it completely
so simple system I can think up would be a system that has only two
configurations
heads or tails according where calling them a table I don't have a corning's
will take
thus our coffee top here like a profound heads
or I could put it down pails about if you can tell a difference from where
you're sitting but I can tell the difference this
this is heads that scales are
so what we have a system in which characterized by two states
two states of being and
we want to add to those two states own law
orlow evolution in going from one instant of time to the next instant a
prime from one be the stroboscopic like
to the next one are what kinda laws are allowable
what kinda laws the basic principles
our physics allow and what kinda blows don't they allow
that's going to be our first kind of question
for tonight up
up
saw first concept is the space
of states in this case just head in payrolls
it just two points heads and tails who 2.0m abstract space
heads entails it's called the phase-space
it's called the facespace up a system space of possible states have a system
and what I mean by 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 day the system will be that's called
buffet space so the system in this case just heads and tails
what kinda laws can you imagine what kind laws of nature can you imagine for
this extremely simple world
it is the simplest well I suppose you could imagine a slightly simple world
you can imagine a world with only one state heads
not much can happen in that world is only one law of physics
heads goes to head since heads Costa heads Coast ahead it's extremely boring
nothing ever happens chris is only one state so how could anything happen
both two states hasn't pails you can have a richer a variety of possible laws
of physics
one law what's pick the berries 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 the time
the next themes in the prime will be heads if it starts tales
it will stay tales that's a very boring law
inlets graph that law by drawing a narrow
if it starts heads then it stays heads
what's just represent that by dre I'm
I'm arrow from head to head and from tail to tail
the meaning of this arrow you start the Palin the arrow
and you follow it NEC the comes around for the same point
that stands for the rule that says
their heads stays our heads and in this case a pale stays a pale sou the very
dole of nature this stays that way for endless amounts of time
and this stays that way for endless amount of time
another possible of nature
would be up
slightly more interesting situation if you look at one instant the time
the next instant it's the opposite heads goes details cause the heads Coast
details
the Czar Peter minister klaus deterministic laws mean
that if you know what is happening at one instant you know for ever 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 there were coming
for app
alright so to draw the graph representing heads Costa pales in pails
goes to heads
we draw an arrow from heads to pails and from tales the heads
and we read that as saying that if you start a heads you go to pale
if you start pails you go to heads what is the
evolution a system under this law of nature you start with heads it heads
tails heads tails heads tails had sales rep rafter
these are two consistent laws over
are physics in a world of only two states
whether we can think of more laws
yes we can but there 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 state is not
very many
we can have a diary buyers and ice
by 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 are weaker have
for example we could simply have with the label these one two three four five
and six
which one is one in which one to six is not very important but they are
but there are six of them we can have a law of physics which is one goes to two
to goes to three 3 goes before pro-gross 255 goes to 16-6 goes back to one
this will be a complicated motion when thought up in terms or a diary
you start with the one up and then it goes to the two up three up and so forth
but it's a relatively simple picture
were drawn in this manner here you can imagine more complicated laws of physics
with six states well with the more complicated or not is in the eye of the
beholder but they are
are first of all we could have a similar law
were instead of one going to to one could go to three
three could go to to to could go to 5
one 23 for
something like that that's really not very different than this
each one goes to a neighboring one well not to neighboring wanna meet when goes
to a next one
and the cycle around and one characteristic
of such systems is able to cycle around for ever and ever and ever
K now you can have more complicated
up I one of the more complicated
different laws for example
you could have a law which goes this way
if you start one you go to go to two then you go to three and then you go
back to one
if you happen to start at three no kissing three this is part right
hard 6666 12
3 for prior if you started 30 to 40 star four you go to 5
back to three in this case well it's somewhat similar to this
you have two disconnected cycles
to do become 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 one to close to three
3 goes to for for all goes back to two
I R 5 goes to section 6 goes back to five
I'm they're all acceptable laws of physics wherever you happen to be
you know exactly where to go next towards deterministic
its deterministic into the future meaning to say where ever you start
you know where you will be arbitrarily into the future
if you start here when you go a hundred thousand primes you just warning that
promotes or 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 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 be
no matter how far you go
that's the character in principle in principle
if not be in their practice some practice things get jumbled up and you
can see them clearly enough for new
you miss detail but as 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 our contest illustrate I
think by drawing some
some possibility use and explain to you why they are not allowed
his law of physics but is not allowed
by the principles of classical physics three states the state is perfectly
alright nothing wrong with that
but let's draw some arrows tomb 3
to goes to three and three goes back to two
I'll be perfectly alright by itself 101
well I could have one goes to warm but I don't want to go to warm
I want to go to
that's completely deterministic into the future or start one
I go to I go to I go to three
ago three back to to back to three
I always know where to go in the next step I just follow the arrow where ever
happens to be
what does it fail as fails to be deterministic into the past
are supposing I know then I'm at
to and where they come from I could have come from one
or I could have come from three so I cannot work my way backward
with unique nurse I can work my way forward
I cannot work my way backward
facts a this is a law of physics switches irreversible
would not allow me to run the laws of physics backward
it would lead to an ambiguity every time I were to
another law which is not allowed by the principles of classical mechanics
or principles a classical physics would be basically the same thing but with the
arrows turned in the opposite direction
are all arrows reversed
here I have a problem not going into the past but I have a problem going into the
future
wet saree arm yup supposing now I find myself for two when I want to go into
the future
I don't know whether to go into the future by following this arrow to warm
on this arrow 23 there are two arrows meaning
out of two over here 1 a.m. goes to three
one of them goes to warm nothing tells me which I wrote to follow
so it's not deterministic into the future I'm I might the
randomly decide to go from 23 a random we 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 while the forbidden by the principles
yup
which one yep
here that one come by with no where to go from three
you get stuck at three yeah
no group
said makin a dangling morning on the way to go next
now you could not say we could try to put something like that in
okay so I would say we go from one to two from 23
the three 2323 now the problem would be in going backward I think
yup yeah here yup right
so one way or the other you get stock car these rules
I can only spot what's allowable what's not allowable was very simple
at every configuration 23
we should have won in and one out we should have a unique one
in unique arrow in any unique arrow out
one arrow in to tell us where we came from
and one haro out to tell us where we're going
that's the character of classical physics
uniqueness into the future unique this into the past when represented in terms
are this very simple
and analog world are analog digital world
ARV are finite number of states
then the rules of physics as we know them
would say every configuration has one
in a row in one hour arrow now
with this role 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 numbers
States always in the same
order you can have slightly more interesting
situations there's no reason why the number States has to be finite
you could have a situation where there's an infinite number of possible states
mistake are responding to every integer positive and negative
are or park or on a position on a line
where the position could be any integer value
and then a simple of Physics would be hugo from one to the next wherever you
are you go to the next warm
each point has one in one hour to can draw them endlessly it'll take forever
by each mark one in worn-out
this of course would also be a rather boring more nature
you just hop from one to the next and the next and next room X for ever and
ever and ever
but at least you wouldn't be cycling around endlessly
again we could have
and some more states are in progress
we could add this on if we start on the line
we simply move often keep going forever and ever and ever
but if we start over here we cycle around
so we could have mixtures are 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 were breaks up into more than one
more than one piece we have something called the Conservation Law
a conservation more is simply a memory
of where we started conservation more 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 I configuration over here with a minus warm
and then we could call we could invent a variable
plus one over here -1 over here it never changes
if it's plus one stays plus one if its -1 it stays -1
that's a conservation law something which doesn't change with time
on the other hand if we hop from plus 12 -1 plus 12 -1
we don't have a conservation law conservation laws
are always associated with these kind of closed
families up 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 they are all mixed together so
there's all kinds are possibilities
that are inherent in these deterministic laws
but always the condition is one in line
in one outline for every point
that can also be called information conservation
its information conservation in exactly the same so that the
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 be
information about where you are is conserved
never changes into the past and the future whereas if you have one of these
bad laws laws
which are forbidden by the rules of classical mechanics than you do lose
information
for example if you find yourself over here
you don't know whether you came from here
or from the ground no it's not quite right trying to stop over here
you don't know where you came from here or here so you lose information
are information conservation is perhaps the most fundamental law
our basic classical physics but you don't lose information about know why
that why is that so
arm it's not written into laws of physics why they are what they are
maybe someday we'll understand hyper loss a physics or metal ores a physics
of people 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 was
a physics fit into this class
I've information conserving laws
even those which a quantum-mechanical
but for our subject this quarter we're only interested in classical physics
are so that's the basic set up if we were interested
only in this stroboscopic world are discrete time intervals
arm the real world of course
is more continuous America
supposing
okay so let's make the law something like this
if the last tomb states where heads heads
then it stays heads
if the last two states were heads tails then it goes two heads
for the last two with heads tails heads then it goes to pails
and I for the last two were tells tales then it goes to pails
I think that serve possible or
a possible law
then you would say i cant tell from the fact that its heads where it goes next
indeed I can't but that would just be another way of saying
that a specification our heads by itself
is not what you would call a state well you would call a state would be the
specification of the previous
last to entries because you need
two entries to tell you what happens next
now that raises the question that that that's very very important classical
mechanics how much
and what exactly do you need to know to say what happens next
if the facespace is the space up things
spacer 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 continue its
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 for continuous-time
is that we need to know the exact location of a particle along the line
we've runner income afraid
barbarians by so let's imagine the motion of a particle online
then you might think that the analog overstate is just a location a particle
where is it
but is it enough to know where a particle is in order to our
to say what happens next now well as the need to know
its velocity in order to know wear particle will be next you need to know
not only where it is
but how fast it moving you need to know is lost city
so that means the state in the same sense that I used it
that what you need to know in order to know what happens next
does not just consist up 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 poor was equivalent to knowing the previous
location
the velocity I'm velocity that tells you
that the facespace myspace are
states the space of configurations is two-dimensional not one-dimensional
it's not just a line its
all online that represents the position of the particle
and a second line which represented velocity
you 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 your NPR Jim
and you have no velocity what if you're over here so we could just
me to draw with nuclear to say that you come back to the same place
what if you over here
you stay the same place because you're moving with zero velocity vertical axis
is velocity see you come back to the same place
what if you're over here
where are you one second later which chopped I'm up into one second intervals
where are you next
somewhere to write her
what if you're over here
know what if you're here
move twice as far of wheat prices far to the right
well if you down here
you move to the left so we could fill up
the space you little tiny arrows to show way you move next
but noticed in no way you move next you have to know not
only where you are in the sense of what X's but you also have to know the
vertical component
mainly with a velocity is so that means the analog
the analog are a point
in the facespace is a point in the space not only have positions but also
velocities
yeah said
yes wipe
good it means
your know it means exactly what the gentleness be 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 or now one way of saying it is
alright in that case I need to know the previous two entries and it just doesn't
want to those questions think or you could say
that the space of configurations doesn't just consist
are a heads or it pales but it consist a pair of entries apparel
yep that's right remember not we model ap
that's where we need two axes is that right we need a an axis
for the present configuration in the past one
yeah attica right
well
of course not because in classical physics the position of a particle and
so forth is a real number
a real numbers so something that you can never determine
exactly are
and so there's always imprecision and that imprecision always represent
a degree ARV on certain p
in 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 a larger and larger time intervals a given degree oven
imprecision
in what you actually know can get magnified in get there are magnified in
to worsen 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 an exactly what they can do
Barack you can always say given a time interval
I want to be able to predict exactly
for the next thirty seconds where every molecule in this room will be
let's forget quantum mechanics right now are I want to be able to predict predict
exactly
where every molecule the will be then there's a certain degree of precision
in the present information 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 I try to
extrapolate for forty seconds I will find the air is get magnified out of
control
if I want to predict correctly for forty seconds I will have to do even better
in my initial conditions in my knowledge exactly what the
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
have to have precision which is so good that way way beyond anything anybody can
do
but in principle given any length of time in classical physics there exist
two degree of precision which allows you to extrapolate for that length of time
the same to the question
you know are soul a.m.
some systems are very predictable some systems are less predictable and get out
of control very quickly the cold chaotic systems
but they are the principles are the same
that that's just the degree of precision the who what you need to know in the
beginning in order to extrapolate for a given length of time
said
well you could I'm not sure what the difference between predictable and it'll
do I use the terms I use them interchangeably so i cant say back
said
the art good
yep card
okay okay yeah
ride what's that
is a problem yeah
right the equations are deterministic
if you know the initial conditions or determine if you are predictable
infinitely predictable if you know the initial conditions with infinite
precision you never do and therefore they never completely predictable
I'm
now me
fact that you need to know both the position and the velocity in classical
fair
physics in order to predict what happens next reflects itself
in the structure of the equations of mechanics
report specifically
it tells you that the equations of motion Newton's equations in this case
Arora call second order equations instead a first-order equations from
illustrated by starting with the first order equation
a first-order equation were first order equation means
is that only has quantities up 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 a complaint accelerations the equations of motion acceleration is the
second derivative a motion
arm weaker right
we know what the real equations of motion and Newton are africa is a main
campaigns acceleration what's right a phony equation
let's ride F equals mass times velocity
force is f velocity is velocity
and let's suppose their force just depends on where you are
we have a particle it moves along a line its subject to forces which very along
the line
I'm force may be big here small there and so forth
so the force the pens on position
and let's imagine this fake equation motion
that's equal to amass times velocity in what is a la city
velocity is the time derivative are the position
the ex by DT
and I will use continuously throughout this course the notation
that time derivative just indicated by a dot
bombings time derivative alright what is coming
well 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 use
just from the equation if I know that the position is a particular position
when I know the force are met and from the equation that tells me what the
velocity years
does it tell me the acceleration how about the acceleration let's see if I
can compute the acceleration
the compute the acceleration from the sick wage in we just differentiate it
once more
we write that be F by DTE
is equal to mass mass is just a constant
primes the second derivative I'll
up the position or just the acceleration
so we have over here acceleration
what about the FTT the time derivative the force
the force varies with time because the position varies with time
I'm so bf by DTE is just a reflection of the fact that the position of the
particle varies with time
and we can write the FBI the TV using standard rules calculus
as just you did the change in half with respect the position
primes a change in position with respect to time
in other words the velocity I we've already figured out what the velocity
years
knowing the force knowing the position so we nor the velocity
and we can read or from the sequester in with the acceleration is
we can figure out all of the derivatives as a motion
if we know where the particle ways by multiplying differentiating this
equation
so what it tells us to make a short story out a bit is it tells us that
if we know where the position of the particle any instant the prime
then we know where it's going to be in the next instant the next to instance
the next three instance
it completely predicts the motion but this is not the character Newton's
equations Newton's equations say F
is equal to mass times acceleration mass times velocity
so let's look at this equation
and because mass times acceleration cannot predict from this what the
velocity is
no there's no equation for the velocity
are if I know what the position is on or before says
because we want the acceleration is but there's nothing in these equations which
to me with the velocity is
that means I have to ed 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
next to a third derivative the fourth the river in all of them
so that tells me then in order to know where I am and where I'm going to be
I have to know position and
the velocity buffet space is a two dimensional space
so we see then that term
that classical mechanics does have this character
are a configurar facespace
are different configurations with offset a 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
heard we will go on and study the classical equations a motion
and study them in a variety of different formulations
but always the
connecting link will always be conservation 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 look
bifurcated
one to
between system but you need to preserve informations there were in your bed
I'm not sure what you mean by replicator
yep
up here he will limit no doubt he me one piece of information to say we'll be
next
if you hear you stay here if you dare you say they're
yup
was dollar is his head entails
the caller is and if you know where you are you know we'll be next
would
cool all the
mineral wherein a is one rule word
good not miss
this is one rule head goes to head scales course details at a single rule
and if you know with your heads then you know you'll be next that heads
if you know your tails you know you so you only need
piece of information has details that doesn't tell you everything
now we are you suggested a different the
well you know
you don't need to know where you were before you only need to know where you
are now to know where you 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 into you come back we follow the
arrow to you get to the next place and that tells you where you be next
you don't need any more information in there
the only question is what wannabes points corresponds to it as one of these
components corresponding
to how much information as a correspond to a point in the space
is it enough to know
heads or tails to know what to do next or Mike you need to know
the two previous things that's different that's a different setup
arm
up
we were truth
we work but stay with the heads and tails for minute wet suppose
the north to say what happens next you need to know with the previous two
configurations
let's do let's work out that example inlets
can write it in this form let's make up a law
I think I had went down before heads heads goes to heads
heads tails goes to work
tales tales heads
because their heads and pales pales
goes to pails ok red
was there something wrong with that
good
all this only depends on your head copyright you're right you're right iraq
Sorry Sorry Sorry Sorry heads yeah we want this one to go to pails
are
good seguimos their heads
ger yup
good word
worked for her part but like what said again
for would
it's also true yeah your your hits in also hard to
work an example for dinner arm let's say
said from switch the second one this one
me details okay I think that's perfectly alright
use information right
well okay if he has it he is a set up
here set up if you know the previous two
mean you know what happens next in fact
I let's let's now see if we can make a table right over Samoa table but sorta
points
the center points then there are four possibilities had ted's heads tails
tales heads and tails tales
okay watching hairs at heads
head pails
hellz heads and pale scales
now supposing you go
from heads heads
two heads some head
heads heads that means you go from heads heads
heads heads right
you from head to head to head to heads now
supposing new goal from let's see what happens if you start with heads tails
where do you go to so you go from heads had pails
to pales pales okay
okay what happens if you go through
tells tales we're gonna run into trouble are we
tells tales Coast details so tells tales goes the tells tales
bad not allowable by the laws of physics
and now trails heads goes where
tells tales Costa tells tales
thread
pails heads here
tales heads coast to coast dead scales serve echoes here
where it seems to me we go we should be able to make a consistent law
good
war water from
whose firm all tells tales goes to our
test has crossed ahead yeah Parker
so then tells tales goes their heads and that means
that tales pails heads so miss tells tales goes to tear it repels heads
are oh okay good good
now we have a workable or now we have a workable law
the only thing we had to do was to say that what we originally called the
configuration
namely a single heads or tails was not a complete
specification and have information a complete specification and information
involved two pieces of information
and once we recognize that we were able to our
to from to write this down
has a law of physics which is deterministic
and reversible okay
so you don't know of here you know not to begin with what pieces of information
you need to know
in order to know what to do next but that's what a state that's what the
configuration space right for the facespace
is it's the collection of all the things you need to know
to know what happened next love course you can 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 simply need more states
to make a year deterministic system
arm what we mean classical mechanics to me more information in the
positions and velocities suppose you need a position velocity is an
acceleration
going through a different equations well would say about the facespace
it would mean you would mean positions velocities and accelerations
are to represent the facespace ok
as it happens that is not the case for
classical mechanics that's an experimental fact
are
part it wouldn't stop us if we did if we did need the accelerations we were just
right third order equations and we will make our phase space three-dimensional
theme
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