Classic mechanics is basically a set of rules about what those laws of motion look like.
It's a set of rule in... in fact, there are two varieties of questions.
The first question is.
I'm not sure about the order of them.
But let's take a first.
The first kind of question is: what are the specific laws for particular kinds of system
Particular system could be a planet moving in the field of um.. of heavy mass.
That has its own particular laws.
Those laws are different than a electrical charged particle,
moving in a field of a magnet.
They're different than a.... you know,
They are specific, specific laws of nature.
Then, there's a more general framework.
What are the rules for allowable laws?
Are there rules for the allowable laws?
What's the grand framework
in which all of the various different specific laws are framed in?
And um...
The second, that's really the second question
which we are really interested in.
The first question will provide us with illustrations.
Illustrations of the principles
that govern what the allowable laws of physics are.
OK, so...
I'd like to start, that I always do in these classes
with a very simple set of illustrations.
Who can guess what my illustrations are gonna be?
Coins, right?
Right, of course.
That's because you've been here before.
Coins.
Let's take a very simple system
that I can think of.
A coin.
I don't have a coin....
So...
When I say a coin,
I mean a abstract coin
lad down on a abstract table.
The only thing about the coin which is relevant to us
is whether it's heads or tails.
The coin has two states.
Two states of being two configurations
or...
two values,
however we want to call it.
Heads or tails.
All right?
So there are two states
This is our dynamical system
It's one coin,
and it has two states
Heads or tails.
that's all there is to the world
to this very simplifed world
Question?
Of course there is an even simpler world.
We can even go back to an even simpler world.
It's a coin with only one side.
Then, it's even simpler:
all it can be
is that one state.
All right? That's a little too simple.
Nothing can happen, nothing ever happens.
So....
This is our world.
What is an initial condition?
An initial condition is it's either heads or tails.
Now what about laws of motion?
Now we gonna make some laws of motion.
The laws of motion um....
of course for this coin that just sit here
the law of motion is very very simple.
Let's um.. let's put two circles here
one stands for heads,
and one stands for tails.
What is the law of motion for this particular coin sitting on this particular table?
The answer is:
It stays the same.
So if we imagine
breaking up time into little successive intervals,
the stroboscopic world
of discrete coin like that
Then the only thing that happen is just nothing happens.
Heads goes to heads.
Tails goes to tails.
Or we can draw a picture.
Heads goes to heads...
Starting with heads,
we go back to heads.
Tails goes to tails.
That's really a very boring law of physics
but it's a very powerful law of physics.
And that tells you however you start,
you know it you will be arbitrarily down the line and time.
All right?
If you start with heads,
the history of the world
will be heads, heads, heads, heads..... forever forever..
So that, that this law may be very boring, but it's very powerful.
Or if you start with tails..
it's gonna be tails, tails, tails...
as far as I can see.
So that's an example of a dynamical system
with a law of motion
where the law of motion is just updating
but it's in this case is very trivial.
There's only really one other law of motion that you can imagine
for this system.
And what is it?
It is...
that whatever is at one instant
at the next instant of the stroboscopic light
it's the opposite.
In the next instant is the opposite.
We can write it this way:
Heads goes to tails,
tails goes to heads.
And we can make a picture of it.
we can picture this law of motion.
by again.. arrows
an arrow from heads to tails
which says that if you start with heads...
The tail end of the arrow always means what you start with
the head end of the arrow means where you end up.
And um...
That would be this law of physics:
Heads goes to tails, tails goes to heads.
Heads goes to tails..
And the history now..
of the world will be:
if you start with heads
you then go to tails.
and you go to heads
and you go to tails
But...
If you start with tails
you go to heads
tails, heads...
Still pretty boring but a little more interesting than the original... first law.
We can write.. we can write some mathematics for this..
in fact we can wirte an equation of motion.
Let's invent a symbol.
A variable
which takes two values.
Let me call it: σ
why do I call it σ?
It's becaue σ is the traditional variable
associated with two values in physics
that has things to do with the spin of an electron
but.. um.. but..
or spin of a particle..
we don't need to know about that now.
σ is variablized either equal to 1
..let's say..
or -1
'1' for heads
'-1' for tails
All right.. so we can instead of calling these heads and tails
we can call it σ=1
and σ=-1
all right? we now have the idea of the configurations space which is labelled by the to possible values
of the certain variable.
It's not as rich as the variable we'll use for example:
positions of particles, values of fields..
or all sorts of other things.
But nevertheless, the mathematical symbol
whose value tells you which of the configuration you are in.
all right, let's take this first law of physics.
The first law of physics...
Not the first law of physics...
the first law that I wrote down
the previous law.
that is heads goes to heads and tails goes to tails.
Let me write that in a mathematical form.
oh, let's call time 't'
In our stroboscopic world
t is an integer.
t is ... 0, 1, 2, 3.. it can also be -1, -2, -3..
so in our stroboscopic world
time is discrete..
and it takes on integer values
so , let's see if we can write an equation
for the bor.. the most boring law in the world.
what it says is that whatever σ is
at a given time, let's put that overhere
σ at time t..
at the next instant t+1
it'll be equal to whatever it was
at the instant t.
this is just the law that says that the spins
or that the coin doesn't flip
but just stay the same.
Whatever it is at one time
it will be the same at the neightboring time.
at the next time.
OK? So that's the simple law heads(5 times)..
Tails(5 times).....
What about the other law?
The one which says the spin...
not the spin...
coin does flip between each um...
between each successive flash of the um...
of the stroboscope.
That's easy.
That's just says σ at t+1
is equal to -σ(t)
If it starts at 1
then at the next instant it's -1
if it start at 1
the next $%&@# it's... it's the opposite.
So there we've written down equation of motion
for a very very simple system.
Notice
that it is compeletely predictive
compeletely deterministic.
There's no ambiguity about what happens arbitrarily far down the future.
Now, of course I can come in and grab the coin and do something with it or do something else with it.
That would disturb it.
In our language that we've set up now
that will be because the system is not closed.
I intervened, and I was not part of the system.
So if a closed system
Laws of systems are compeletely deterministic in classical mechanics.
OK, now let's go... let's think abouot
a more complicated system.
More complicated system..
not much more complicated
Intead of a coin
Let's take... what's the next case?
A die~
A die.. half.. half of a pair of dice...
You know I never knew until I start teaching this that the sigular of dice was die..
I feeling good(is this right?)
Somebody corrected me once
I call it a...
I guess I call it a dice..
simply meaning a die
I guess I did.
OK, so we have a die.
A die has 6 states.
1, 2, 3, 4, 5, 6.
That's the only significance um...
for our purpose today about the fact that it's a die..
is that it has 6 states.
Again, we could label the six states: 1 through 6
Let's put them 1, 2, 3, 4, 5, 6.
And these stand for the 6 different configurations of laying the dice...
the die on the table.
That's the initial condition.
And well sorry, the initial condition
is a choice of one of these 6 configurations.
What about the law of physics?
What about a law of motion?
An equation.. not an equation..
I am not gonna write an equation.. it's little bit too complicated..
but what's the possible law of physics
for this simple system?
Well, a very simple law would be:
I'll give you a very simple law..
the simple law is nothing happens.
Nothing happens however you start you stay the same way.
To graph it, we will just graph it like this.
Whatever you start with, it's what you get in the end.
That's supposed as boring as um... as the um...
as the simple coin.
OK, a more interesting law..
which will be you cycle around these..
collection of configurations.
For example
A possible law of physics
would be 1 goes to 2
2 goes to 3
3 goes to 4
4 goes to 5
5 goes to 6
6 goes back to 1.
I leave it to you to try to write an equation for this.
An equation of motion.
That's.. not hard to do.
I'll just leave to you to do.
OK, but you can see what happens.
The history of the world would now be, well..
give me the starting point: 4.
After that, the history of the world is:
4, 5, 6, 1 , 2, 3, 4, 5...
endlessly cycling around.
And that would be
the theory of this particular die
with a particular equation
or a particular law of evolution.
Now we can write down other laws.
other laws, and the simplest way to write them down is to graph them.
Oh again, notice that it's compeletly deterministic.
Another law would be to cycle in the opposite direction.
2 goes to 1, 1 goes to 6, 6 goes to 5..
Or, we can do more complicated things that not really more complicated..
that just more complicated to draw.
1 goes to 2
2 goes to 5
5 goes to 3
3 goes to 4
4 goes to 6
and 6 goes to 1.
Now, this law is logically equivalent
to the one which just cycled around.
It's just a relabeling.
It's a relabeling of the states.
But it's again
one cycle.
1 to 2, 2 to 5, 5 to 3, 3 to 4, 4 to 6, 6 back to 1
If I were just rearrange them.
redraw them
then it just looks exactly the same as the original cycle.
So this law would be..
what should we call..
logically equivalent to one and the others.
OK, now let's think about some laws which would not be logically equivalent.
Here is a law that's not logically equivalent.
Again, six...six...um..points.
I won't bother labeling them with numbers.
1 goes to 2.
2 goes to 3.
3 goes back to 1.
4 goes to 5
5 goes to.. sorry 6 goes to 5
5 goes to 4
4 goes back to 6.
It's again completely determinisitic
wherever you begin
the future is laid out for you completely.
If you start with 3
you go to 1 to 2 to 3..
On the other hand, if you start with 4
you go to.. oops...
you go to 6, 6 to 5, 5 back to 4 and so forth.
so it's again, completely deterministic.
But it's not logically equivalent to any of the previous ones.
It now has two cycles.
And if you are in one of the cycles
you will never get to the... I'm giving them a name now
cycles, ok?
if you are on one of the cycles
you will never get to the other one.
So....
It's not that there are 2 systems.
It's one die
only one die
but with this particular law of physics
you are trapped on one cycle or the other cycle
There's a name for this kind of behavior.
It's called having a conserved quantity.
A conserved quantity is something non-trivial
that you can label the system with
which doesn't change with time
For example, we could label this cycle up here
cycle NO.1
and assign it a value 1
the cycle down here
we could label a value '2'
and then we'll say that if you start
with '1'
the quatity '1', or the kind of conserved quantity '1'
you stay with '1' forever.
If you stay with '2'
if you start with '2', you stay with '2' forever.
This is called a conserved quantity.
In other words
A quantitative quantity or a quantity that is quantitative
but a quantitative quantity which just doesn't change.
It's conserved.
Um...
Let's call it the conservation law.
So.., this cycle over here, the single cycle system
doesn't have a non-trivial conserved quantity.
You pass through every one of the states.
This has a conserved quantity
Either '1'.. what did I say '1' or '0'? I don't remember what I signed them
but I signed them 2 different numbers this one and this one.
Or we can make up more complicated, don't on a serially more complicated but other examples.
Alright, this one... 1 goes to 1.
If the die is at 1, the rule is it stays at 1.
If it's at 2
it goes to 3
and if it's at 3
it goes back to 2.
and, if it's at 4
it goes to 5
if it's at 5, it goes to 6, if it's at 6, it goes back to 1.
I've changed my color coding, sorry about that.
Alright.
Again
if there are 3 cycles now
there 3 cycles
and if you get on any one of them
if you start on any one of them you stay on it.
You could describe this
by saying the first cycle over here
correspond to the value 0 let's say.
correspond to the conserved quantity that having value 0
it could be '1' over here
and it could be '2' over here.
If you start with a conserved quantity being '1'
then it stays '1'
if you start.. what..what I say...sorry..
'0'
if I start with a conserved quantity being 0, it stays 0
If I start with being '1'
that doesn't tell me exactly where I start
either here or here
but, it tells me that I start within one of these two.
and I stay with that value.
I don't change.. I will change the state
but I don't change the value of conserved quantity
Likewise overhere.
So we can consider a variety of different.. logically different..um..
evolutions
what is it that distinguish them?
Well, really is just the number of cycles.
The number of distinct cycles.
This is three.. um...no... um..
It's a little more than that
it's a little more than that
but..um, but you can see, you can..
you can play around with this
and investigate it
and you'll very quickly get the point
if you haven't already gotten it.
Any question up to now?
Yes sir.
Student: In the very first example you talked about coins
where you have projections either T T T or H H H or H T H T
So..um.. those, I mean I can imagine that the of H and T.
Yeah.
But..um..but if I would try to construct one of those
I think of it have to depend on more than just the preceding state meaning several states factors.
Leonard: Not for that..not for those laws
Student: Um... HTTHTT..
Leonard:That's a different...right..right..that's right
You could invent laws
where to know what happens next
you have to know not only the previous
but the second previous state.
Right, that's correct.
Student: Is that aspect import for sure?
Leonard: Yes, yes it'll become important... it'll become important.
But under those circumstances, you would say..
that the state of the system is not characterized
by just whether it's heads or tails.
It's characterized by..
the configuration and the previous configuration.
And so you'll have, in that case
4 configurations
The 4 configurations would be:
heads preceded by tails
heads preceded by heads
tails preceded by... and then you will...
and then you will do the exactly the same thing
except, you'll say there are 4 possible states.
Right..so...and
that of course is that
I know exactly where you want me to go
but I am not go there yet.
But...um.. right...not yet.
So, it's what I said before
the state of a system
or the configuration consists of all the things you need to know to predict the future.
So in the case you described
what you need to do..or what you need to know
the initial condition consists not only of what the coin is doing
but what was last doing.
Good.
So that's a good point.
And um...
We'll..we'll come back to it.
All right so we have the idea of configurations
we have the idea of isolated systems
we have the idea of conservation laws
Let's just point out
that there's nothing to prevent us from considering systems with infinite number of states
it doesn't have to have infinite number of objects
to have infinite number of states
it just needs to have one object
which can..um..be rearranged in infinite number of ways.
Um...
For example:
If we had an infinite line
and on that infinite line we marked off the integers
and we set this as particle
what does a particle mean?
A particle now just means
a thing which occupies one of the places
the places now consist only of the integers.
what's in between doesn't matter doesn't count
All right.
So, a particle is simply an object
which sit at one of these integers.
You can have an infinte array like this
and then you would say the particle has an infinite number of states
namely, the infinite number of possibilities of where it can be.
We can label them by an integer 'n'.
And then we can invent laws
such as...
wherever you are
move to the right 1 unit.
wherever you are move to the right 1 unit
and that will be a picture looks like this.
It doesn't cycle
Nevertheless, we going to call such a thing a cycle.
But...um..but..
you can see, it dosen't cycle.
It just goes on and on forever and forever.
We could have another law
the other law could be wherever you are
jump 2 units ahead.
Wherever you are....jump 2 units ahead.
Um....no...I think I'll do like this way.
IF you're overhere jump 2 units ahead
if you're over here, jump 2 units ahead
if you're overhere... if you're here, and so forth...
All right? So we make a picture then of a kind of...
law of physics which tells us to jump 2 units. Again..
Make up an equation for that.
Make up an equation for that..
Thaaat's easy to do..
In the first case
there was no interesting conservation law.
There was only one cycle if you like...
Um...wherever you are
you alwways either
get to any other points
or if you go back into the past
if you imagine running you back to into the past
you'll have come from that point.
So..so there cannot be any interesting conserved quantity
because you passed through every single one of these integers
and so they can't be distinguished by a conserved quantity. In the 2nd case..
There is a conserved quantity
and you can call it the 'oddness' or 'evenness'
of... the position of the particle.
You could give all odd numbers
or represent them by a value of quantity whch is '0'
and all even...that's..that's weird isn't it?
Let's make the odd numbers have an integer number '1'
and the even numbers be labelled by integer number '0'
and then you would say there is conserved quantity which can either be
'0' or '1'
If you start on one of them
it stays there
if you start the other
it stays there..
All right. So having an infinite number of umm... of configurations doesn't change the picture very much
It does open the possibility that
there's a kind of endless evolution which never repeat itself.
So in that sense, it get a little more intersting.
And you can think of a sort of generalizations of this.
All right, so I laid out some... I call them allowable laws of physics.
Allowable rules.
Let me talk now about some rules which are not allowable.
Now, not allowable by who?
By me....(OTL..)
But of course, I'm simply reflecting the way..
classical ..real classical physics works when I say allowable and not allowable
Um....I'm giving you a...let me draw a picture of an non-allowable
or unallowed law
it's completely deterministic
it completely predict the future.
But there are something different about it
than the laws I've drawn over here.
I am gonna do it by drawing a picture for it.
Let's got...we can make up many many examples
but for this particular example let's get 3 states
so this is a 3-sided coin.
A 3-sided coin let's get heads tails and....
I don't know..
Size....Heads, tails
I'll just gonna fix something little more inscape ... but...
tails..tails is about inscape as what we gonna get.
Yeah edge.
Saw edge, that's pretty 'edgy'.
Heads, tails and edges.
OK.
all right, here's a law of a kind
which represent something that we'll not allow in classical physics.
Heads goes to tails
tails goes to edges
and edges goes back to tails.
All right. Now wherever you start
wherever you start, the p... the.. history is completely predictive.
If I start with tails, I go to h.. I go to edges.
Tails edges, edge tail..tail..so goes tails edge..tails edge...
just follow the lines.
If you start with heads
you go to tails
then edge then tails
then edge, then tails.
So here is vari.. some histories
First of all, starting with heads
Heads tails edges tails edges tails ...
If I start at tails
Tails edges tails edges..
If I start at edges....
Now what is it that's odd about this law?
What's odd about this law
is that it's completely predictive if into the future
But, it is not predictive into the past sort of speak.
Um...
If you know that you're at tails
you don't know where you came from.
Did you come from edges?
Well, maybe.
If...but you could also come from heads.
So, while you can predict the future
you can't....redict?
what's the opposite of predict?
Retrodict! That's what I'm looking for.
You can't retrodict the past from this law of motion.
One configuration or several configurations
run into the same configuration
And so you can't tell if you're over here whether you came over here or here.
The word for this is that it's not reversible.
Here's a way to think about it.
Reverse every arrow
Now you have an unpredicted situation.
If you're tails
you don't know whether to go to heads or whether to go to edges.
So.. it's a predictive situation one way
but not retrodictive the other way
the word for this is not reversible.
I won't call it irreversible, let's a little too definite
It's not reversible.
You can't...
you can go one way but not the other way.
This is the kind of law that is not allowed in classical physics. Yes sir, oh madam... I can't see...
Is classical allow that if the probability with time in each possibility?
Well..um..
OK.. that of course depends on what you trying to represent.
Um....
Classical physics doesn't allow probability
so I think I can escape from the question by just saying:
we don't do that in classical physics.
But..um..
We could..we can imagine ..
Cuz we can imagine everything we want
In fact we can imagine this
and we can study its properties
The point is in one way or another
it conflicts with the rules of classical mechanics.
Probability or let's call it non-deterministic laws also conflict with the rules of classical mechanics
So...
It's a very good question
and it's something we'll come back to.
whether when..when..quantum mechanics for example
which is not determinisitc
It's quantum mehanics have a analog of this reversibility idea
even though it's not deterministic.
And the answer is 'yes'.
but not tonight.
Student: classical statistical mechanics..
Leonard: Well, classical statistical mechanics also has um..
keep in mind that the rule of classical statisitical mechanics
begin with the laws of mechanics
ok?
We begin by assuming..all right..
let's talk a little bit about the limits on predictibility for a moment.
If we have a perfectly predictive system of equations
it won't allow us to be completely predictive
if we don't know the initial condition exactly.
So we need to know two things to predict the future
one is what the rules are
the laws..
and one is what the um... what the initial conditions are.
Now, in these very very simple systems
it's easy to imagine that we could know exactly what the initial conditions are
Of course we may not.
We may know that we begin with these.. not this one..but..
this one.
we may know we begin either with edges or heads
we don't know which.
And there will be some ambiguity in what happens
no ambiguity because the equations have ambiguity in them
but because of knowledge of the initial state is inperfect.
Ok? It's easy to understand.
For these simple discrete systems
It's easy to imagine that we can do enough experiments um..
very quickly
just look at the coin
and know the initial condition perfectly.
In the real world
where we face with degrees of freedom which are continuous
meaning to say they can be any number on the continuous real axis
any number of numbers
the positions and velocities of all the particles in the world
You can never know them perfectly.
Um....
No matter how many decimal points you may have
that you may have measured
decimal places you may have measured about the location of a particle
you still don't know it exactly.
So, there's always a degree of ambiguity in your knowledge of initial conditions.
that degree of ambiguity may or may not
sort of blow up in your fates
that's more changes
in what the initial conditions are
may or may not give rise to large changes in what happens in the future
So the right way to say it is
if you knew...
we have.. we have to be quantitative
but if you knew the initial conditions perfectly
then you can predict the future forever and forever
and that's a true classical system.
And you could also predict the past perfectly.
If you have inperfect knowledge of the initial conditions
you want to quantify that
how inperfect is it?
And if you can quantify it
it may allows you to answer the question
how long into the future can you predict things.
So if you knew enough about the initial conditions of the atmosphere
you might be able to predict the weather for 3 days.
But, because your knowledge is limited
and because the atmosphere is one of the systems where little errors build up
they call it chaotic.
Because the atmosphere is chaotic
no matter how well you know the initial conditions
it's always true that if you wait long enough
you won't be able to predict the future.
On the other hand
if you say I want to be able to predict the future for x number of years
it should be possible to say how precisely you have to know the initial conditions.
OK!
So...um..when we come to the real world
this idea of predictability becomes a little more complicated
But I started on purpose with very..um simple discrete systems.
OK, so...
Laws that are allowed
Laws that are not allowed
Laws of...um...classical mechanics
are not only deterministic into the future
but they are deterministic into the past
which makes them reversible.
Now, how do you look at one of these pictures
decide whether it's a legal law or not.
Well, it's very simple.
If every state has one outgoing and one incoming arrow
that means that you know where it came from
and where it's going
So, when you draw one of these pictures
if you want to know whether it is an allowable legal law
in the sense that I've defined
I'm using 'legal' now just as a..a term for a...for...
reversible systems..
If you want to know whether it's reversible
look at the arrows
and if each state has one incoming and only one incoming
and one outgoing only one outgoing
then, it is both deterministic and reversible.
There are analogs of all things I'm telling you now
about more complex systems or more interesting systems
such as particle moving around.
OK, so that's um...that's sort of warm up preliminary about what classical mechanics is about.
It's about predicting the future
or using the predictability the fact and principle that you could predict the future
in other ways such as statistical ways
to..um...to...limit
what will happen.
All right, now we want to move on
to a more realistic world
and in fact the world of particles
we gonna be interested in world of particles moving... and a particle for our purpose can be thought of as a point particle
if we want to make a more complicated system
we'll make systems of particles, points particles.
But, we'll consider point particles moving around in space.
So...
What do we have to know
what are the configurations of a point particle?
What do we need to know
um...well..ok
before we do so
I think we should do a little bit mathematical preliminary.
I want to remind you for those who don't know
Or who know..knew but don't remember
or remember but just barely.
What vectors are and what coordinate frames are.
Coordinate system is just a way of describing space
quantitatively.
And incidentally, for our purposes
today and largely in general
we'll of course assume that the space is 3-dimensional
and so a point of space will be labelled by 3 coordinates.
But we're perfectly free to think about systems which are higher dimensional or lower dimensional
and we'll do so
since we are interested in formulating the basic principles
we don't have to restrict ourselves to very very specific examples
a particle could move in one direct.. in one dimension.
it can move in 5 dimensions
and...and we'll be interested in all the possibilities.
but for the moment
let's just think of a particle as thing whch moves in 3 dimensions.
All right. So in order to be quantative
about the location of the particle
we introduce coordinate system
Coordinate system, Cartesian coordinates will be the usual things we introduce
Later on we'll describe other ways of describing location of particles.
But for the moment, we take space
we identify an origin.
The origin is um....
up to us... where we want to put it
I can put it over here
I can put it over here.
It should be that the important question that we're interested should not depend on the convention of where we put the origin
But it's useful to fix it once for all.
and say the origin of the coordinate is located
in .... whatever.
Wherever we want to put it.
Here it is.
Then introduce axises.
The axises are taken to be mutually perpendicular
you can check that they are perpendicular by um..by..
t squre or whatever it is that you use to align axises
and you label them
you can label them x, y and z
or x1, x2 and x3
we use both kinds of notations.
But there's also an ambiguity about the orientations of the axises
give that they are mutually perpendicular
we still have to decide..you know..
I don't have to draw..
you know what I mean.
Which directions they going
and so, that's like fixing the origin of coordinates we also have to fix an orientations for the x and y axises
once we've fixed the orientations for the x and y axises
the third one is fixed, it's perpendicular to it.
Incidentally, there's a convention
and the convention is called: a right-handed coordinate system
the convention is:
when you've picked x and y
you still need to know
one discrete piece of information
is z pointing out of the blackboard or is it pointing into the black board?
and.. we settle on that by a rule
it's a convention, it's arbitrary the right hand rule.
If we take our thumb
and our index finger
thumb along x and index finger along y
and z is the middle finger
the direction of middle finger.
All right. So that's the right hand rule
And selects out
this coordinate system from the other one
with z goes in the other direction
OK.
So that's the idea of a right-handed coordinate system
we also have to mark off distances along here.
So distances are marked off..
equal distances
with a ruler
and of course
in saying that we mark off distances
we are also assumimng a..
set of units
the units could be meters could be inches, could be feet, it could be light years.
so, again another not ambiguity but another convention
is to choose on units
but once we chose on units
we can lay off distances along here
and then every point in space
can be labelled by the values of x, y and z.
It has a height: y
It has an x, x and it has a z, z
If you like you can think about how do you get to the point from the origin
you go certain number of steps along x
you go certain number of steps along y
and you go certain number of steps along z
and those quantities x, y and z
are the coordinates of the point.
So a point is labelled by a set
now I know you all know this.
But let's spell it out anyway.
OK? So, that's the way we describe a point, that's the way we describe particle.
And of course if we have a system of particles, many particles
then of course we just have to put in
one such point for each particle
ok
Vectors. What is a vector?
A vector is an object which has both length and direction.
For example, a very simple....vector
is the position of this point relative to the origin
Here's the origin
and the position of this point relative to the origin has magnitude
which is the distance from the origin
and it has a direction
Namely just the direction of the...of the arrow connecting the point with here.
Think of that vector
As an object which has a length and the...
direction
but don't think of this being located anywheres
Think of it as...
being the same..
no matter where I draw in space
In fact I don't even have to think about drawing it in space.
It is what it is.
It's a magnitude and a direction in space.
That's called a vector.
And from now on
we will label vectors
by...
putting a little bar on top of them.
If I really conscientious I'll put a little arrow on top of them.
If I forget the top of the arrow
I get bored writing arrows
I'll just put little bar on top
And here and there
I may even forget to put anything on top
but you..you remind me.
OK, so a vector has a magnitude
the magnitude is its length.
It does not have to necessarily be ..
a relative position.
It could be a velocity, it could be acceleration
Something that....or... other things
It could be...
an electric field
be all sort of things
The criteria for to be a vector
is that it has a length
and a direction
OK, so that's um...that's the notion of vector.
Every vector
can be described
or..or has associated with it a length
in fact for the length of it I don't have to put a vector sign
Let's call the length of it..
Two bars on either side of it.
The absolute values of it are called its length
This is the way of writing its length.
So that's the length of a vector.
And a vector, it's always possitive or zero..
it could be zero
if the vector has no length at all
In other words, if this point is right at the origin will has not length at all
It's always either possitive or zero
So every vector has a length
And it has a direction which is not so easy to write down.
OK. It can also be described by components.
A component, the components of the vector are exactly what I said before
If you want to go from the origin to this point over here
you'll go certain number of units of x
certain number of units of y
certain number of units of z
the other way to define it
is to drop a perpendicular..
from the point to each one of the axis
I..I don't draw this very well.
Um..let's see.
I think...how we gonna do this..
I want to get this vector to be out of the black board.
how can I get it to be out of the black board by drawing it on the black board?
I mean the time I draw it on the black board it'll looks like it's on the black board
OK.
Here to here.
OK.
We drop a perpendicular
from the tip of the arrow to each one of the axis
And that laids off for us
the distance along those axises
which are the three components of the vector
we'll call them
v_x, v_y and v_z
when I'm writing the component I have no need to put the arrow on top of them
the components themselves are numbers
OK?
What is the length of this vector who's components are... v_x, v_y and v_z?
Yeah... the sqare of the lenght is v_x^2+v_y^2+v_z^2
Pathygoras's thorem except in 3 dimensions
I'll assume it, we won't prove it.
Umm.. the length of v
the magnitude of v
is the sqare root
of the sum of the components
That's the notion of the magnitude of a vector
the direction of the vector
and the..um...
is encoded in the ratios of different components
For example if the x component is much bigger than the y component
then the vector is pointing more or less along the x axis and so forth
the components can be possitive or negative
if they're negative, they pointing along the negative axises and so forth
All right, but as I..as I
I want to emphasize
now..now...again
that the notion of a vector
is not necessarily tied to the location of anything in space
it is what it is
it's a length or..or magnitude
and a direction
and if you move around
it doesn't change the vector
this is the mathematical definition of what you think about vectors
you don't think of as being tied to a point in space
you can...I mean there are circumstances that you may want to tie ..um...
a vector to a point of space
for example, you might want to ask:
what is the electric field at a particular point of space
then that vector is tied to that particular point of space
But...the notion of vector transients that:
doesn't matter where you put it
All right, now we have to talk about..the algebra of vectors
I really feel sorry for all of you people who
were said the same endless number of times before
But...um...
It's funny
like a book
which...no matter how good the book is
pretty much, except for some extremely special cases
you really only want to read it once
you know...you may say you want to read that book
you love that book so much you can read it endlessly
but it's not really true you read it.. once
Um...there are other things like good piece of music
which you want to listen over and over and..
it doesn't matter how many times you heard it
it just as always good to hear
I assume my lectures are like that
What? You're kidding me..
All right.. so let's talk about the algebra of the vectors
The algebra of the vectors are first of all:
you can add them
you can substract them
you can multiply them by numbers
So let's talk about that a little bit.
Given a vector
Again, that's a length, and that's a direction
Let's call it vector A
you can mutiply it by a number
an ordinary number
Not integer but a real number
Let's mutiply it by the number c
what's the definition of that depends on whether c is positive or negative
if c is positive..
it's a vector along the same direction
except it's length is multiplied by c
so twice of a vector
means a vector with an exactly the same direction but twice as long
and so forth and so on
we can also define
the negative of a vector
In other words we can...let c be -1
we can either put here -1
or just minus the vector
and of course, the negative of a vector is exactly the same vector except with the direction
in the opposite direction
That's the definition of multiplying a vector by a number
by an ordinary number
and you can also add vectors
that's just remind us about the soul of rules for adding vectors
there are 3 ways of thinking of adding vectors
and the first way...
let's say, A+B
We take A and lay it out
here is..here is one end of it
tail end of it.. and
here is the arrow end of it
and we take B
and we put the tail of B... at the head of A
we draw a triangle
a triangle has its third leg C
which is the sum of A+B
it might be a degenerate triangle
it might be that B is along the same direction as A
In which case C will also along the same direction
and looks something like this
In which case, it's kind of..um
degenerated triangle
But still, we just think of it as an triangle
we can add vectors by this rule
we can multiply that vector by numbers
Oh.... before I do that
Let's just talk about two other ways of adding vectors
That are trivial
In according to this rule
It looks like it might depend on which one I lay down first
But of course it doesn't
you can choose a rule
which is symmetric between two of them
if this is A and this is B
you put the tails of A and B down together
you draw a parallelogram
and you draw...
the diagonal of the parallelogram
that's callled C
and this form...it's completely manifestly clear that it's doesn't matter
whether you put A down first or B down first
so A+B is the same as B+A
OK, so that's um... that's the vector addition
you can multiply vectors by numbers either possitive or negative
let's call this... a
we know exactly what that means
we can....multiply B..
by a number b
And we can add them.
We know how to multiply by numbers
we know how to add..
and that means we know how to construct the vector
a times vector A plus b times vector B
we know exactly what that means
OK?
The third way of adding vectors is to use the components
So if we have two vectors with components
Ax, Ay and Az
and Bx, By and Bz
then....Cx is just equal to Ax+Bx
and likewise: Cy and Cz
we can summarize these all
by saying C sub i
where i could be x, y or z or 1, 2 or 3
is equal to Ai+Bi
and that's an equivalent to a useful way to add vectors
since often we specify vectors by the um... by their components
all right, if we can add vectors
and...um...
subtract vectors and multiply them by numbers
the natural question is:
can we multiply vectors?
can we divide vectors..
well, no, we not gonna diveide vectors
the idea of vector division is not a well defined idea
Um..but...
Vector products...
There are two kinds of vector products
two distinct concept of multiplying vectors
In one of them, when we multiply two vectors
we don't get another vector
we get a number
That's thing is called a scalar
sometimes a number is just called a scalar
so...one....definition of the product of two vectors is called the dot product
let's talk about the dot product of two vectors
A dot B
A dot B is defined in the following way:
You take A
you take B
and now you think of the component of A along the axis of B
what is that mean?
that means you drop a perpendicular
from the end of A
to the axis, not to the x axis not to the y axis not to the z axis
but to the axis defined by B
So you drop a perpendicular...
and now you take the length of...
let's call this here...
"A sub B"
the component of A along the B axis
you take...the component of A along the B axis
and you multiply it by the component of B along the B axis
what's the component of B along the B axis?
It's the magnitude of B
all right?
so if you take the component of A along the B axis
you multiply it by the component of B along the B axis
and that's called the dot product
all right?
I'll write it out in a minute more definitely
That's called the dot product.
But the way defined it
it looks like A·B might not be the same as B·A
after all
the rule for B·A would be
to drop a perpendicular from B onto the axis defined by A which is a different thing
and then multiply them together
is it obvious that A·B and B·A are the same thing?
Well, if we can write them in a manner which um...
is symmetric between the two of them
then we will know that they are the same thing
OK, so let's call this angle here θ
The magnitude of A
is the length of A
the magnitude of B is the length of B
what is the component of A along the B axis?
A_B is equal to the magnitude of A times the cosine of the angle between them right?
let's call it θAB
θ is an angle now
and that's the angle between A and B
That's the component of A along the B axis
We now multiply that by the magnitude of B to get the dot product
all right? So the dot product is A·B
and it's equal to ABcos(θ)
I won't bother writing θAB
A along... the component of A along the B axis
that's this..
time the component of B along the B axis
which is just the magnitude of B
it's AB...magnitudes of A B cos(θ)
now and this form..
it's completely clear that it's completely symmetric between A and B
and it doesn't matter in which order you multiply them together
the cosine of the angle is just the cosine of the angle
is A·B always possitive?
and under what circumstances if it's not...under what circumstances it would be negative
if the cosine is negative!
when does the cosine is negative?
the cosine is negative if the angle is bigger than 90 degrees
so....that would be the situation for example...
where A is pointing like that
then the component of A and B direction would be negative
cosine of θ would be negative for an angle bigger than 90 degrees
Um....the...
cosine is negative..
what about two perpendicular vectors?
zero!
cosine of θ is zero for 90 degrees
so a diagnostic
for deciding whether 2 vectors are perpendicular or not
is the to calculate the dot product.
Now. That can be useful
the reason it can be useful is becaue we can express the dot product
in component form
If we express the dot product in conform... in component form
I won't prove this..
it's not hard to prove
that in terms of the components of vectors
this is equal to AxBx+AyBy+AzBz
in other words, you multiply the components and you add
that can be easily proved little bit of...
little bit of trigonometry not much
um....that...this is equal to
this over here..
that's nice, let me show you how can you use that
Let's see... do we really need that?
Well, if I just were to give you
the components of A 3 numbers
and the components of B 3 numbers
and ask you to compute the angle between A and B
you first reaction might be to throw up your hands
and says well I don't know get me a protractor..I'll try to..I'll try to measure it..
But...here's a tool now
you calculate the dot product between them
that gives you A·B
you can also calculate...
what is the magnitude of A?
It is the dot product of A with itself
Let's write A...oh sorry that's not quite right
A dot A, what is that?
That's just the square of magnitude of A
so that's A squared
Likewise for B
So...you could calculate the magnitude of A
or the square of A that's x^2+y^2+z^2
likewise for the magnitude of B
and then you'll calculate the angle between them
by calcualting the dot product of two of the vectors
so from the magnitude of each one of them
the dot product of two of them
you will compute the cosine of the angle between them
all right? so that's a good trick..um... if you want to know the two angle between two vectors
Let's prove..let's prove the law of cosines
let's prove the law of cosines...just a simple elementary thing
supposing we have two vectors A and B
and we want....think of it as two sides of a triangle
and we want to compute the length of the other side of the triangle
so let's call that C
C=A+B
how we compute the length of C?
is just C is equal to A+B squared
or A+B dotted with A+B
the square of the lengthe of a vector
is the dot product of the vector with itself
so if the vector happens to be C
that's C dot C
that's the square of the length of C
I am gonna make A....um.. sorry
A minus B...A minus B thank you
C is A minus B
it's not A plus B
A plus B will be over here
we could do it that way couldn't we?
let's .. do it A-B
and you can work out very quickly that this is A-B do it with yourself
OK, so it's A-B squared, what is it equal to?
This is equal to: A·A+B·B-2A·B
All right? Just ... multiply them together
and this is just square of A
This is A... magnitude of A squared plus the magnitude of B squared
and this is minus twice the magnitude of A times the magnitude of B times cosine of the angle between them
That's called law of cosines
That the size of the third...um...
leg of a triangle is given by the sum of the squares minus twice the product of the lengths
times the cosine of the angle between them
ok so that's...that's vector addition versus vector multoplication
vector substraction...
even vector multiplication the dot product...we'll...we'll worry about the cross product
another...another time not today
Um....all right now..more...not more important
but... more interestingly..we have not only a algebra of vectors
which means adding subtracting and so forth
oh, as I said, there's no notion of division of two vectors.
Um....
I notice that for the dot product
The product of 2 vectors is not a vector
It's a number
The cross product is another matter
that we'll come to it another time
OK, now..um....let's talk first of all
about a particular vector
which characterizes the positions of particle
Here's the position of a particle
now think of this as a particle
in fact we gonna allow it to move around a little while
and...the origin is a particular special point that we've picked out
we do have to worry about what'll happen if we change the origin
but we not gonna do that today
and therefore the position of the particle defines a vector
that vector is usually called little "r"
r is supposed stands of radius
and in fact, the magnitude of the r
is the radial distance from the origin to the particle
that is true
um...presumably..
I don't know where the notation came from
I think it is from 'radius'
but, for now, it stands for the position of the particle
what are the components of r?
They are just the x, y and z of the particle
they are just the coordinates of the particle
x, y and z
so r has coordinates x, y and z
of...forming that right that way
r sub x
is equal to x
the position x
and so forth and so on
That's the notion, the simple notion of the location of a particle
described as a vector
now....
in general
we are interested in the motion of particles
the motion of particles....is what classical mechanics is about
how the motion goes from one instance to the next
in other words it's how it update from one instance to the next
and so if you think about r as a function of time in general
It moves around
however moves around
we assume it moves around continuously
differentiably
smoothly.
at the same time, its components
if r is a function of time
then r sub x
is also a function of time
likewise for y..y of t
and z of t...
and so the motion of particle is summarized in this case
by three functions of time
x, y and z.
what about the velocity of a particle
the velocity of a particle is also a vector
it has a direction.
It's non-necessarily that the direction of the position of the particle
from the origin
for example
the particle might be overhere
but moving this way
moving out of the black board
in that case....its velocity will be this way
but its position will be this way
and so they are two separated distinct vectors
the position could be any vector
and the velocity, any other vector.
But how do we describe the velocity?
The velocity is the time derivative of the position
all right, I am not going um...spend a lot of time explaining that fact
I think you all probably know..
Not...um..you're probably in a little over your heads
but...I'll assume you know this
That..the velocity of a particle
is the time derivative of the position of the position of the particle
so we can write that in a number of ways
we can first of all
say that the components of the velocity
the velocity along the x axis is the derivative with respect to time
of the position... of the x component of the position
or we can write it either as r sub x
or we can just write it as dx/dt
and likewise
for the other two components of velocity
so the velocity is also a thing with three components
it's also a thing with length
the magnitude of the velocity which is called the speed
the magnitude of the velocity..
and it has a direction
and...the components of velocity
are just derivative of components of the position
so, there's a velocity vector
and it's easist to specify
by specifying its components
dx.....
I won't write equals..
it ha...it has components ... x y and z
which are dx/dt, dy/dt, and dz/dt
that's the notion of the velocity vector
and..I will frequently and almost always
use the notation which many of you know
but... I introduce here anyway
Differentiating anything with respect to time
calculating its rate of change as time preceeds
There's a traditional notation for it
some of you know it, some of you don't
I'll tell you right now
that the derivative of anything
let's call it any function of time
with respect to time...
is just labelled
this is just have to...this is just in order to keep from having to write d by dt over and over..
is just..put a little dot on top of it.
dot...
definition of a dot on top of a function
is its derivative with respect to time
it doesn't mean the general ....
you wouldn't use it for derivative with respect to space
you wouldn't use it for derivative with respect to anything else
dot means derivative with respect to time
so, the...we can rewrite this saying, that the component of velocity
let's write it in one formula
v sub i
i could be x,y or z
is equal
first of all to dx sub i by dt
which is the same as x sub i dot
that's the notation for velocity..
velocities are important
let's..um work on an example
well, no, before we work on an example...
let's talk about accelaration
what is acceleration?
is the rate of which the velocity is changing
acceleration is 0
if velocity is not changing
whenever the velocity changes this is the acceleration
acceleration is also a vector
it's not just 30 miles an hour per hour
or 30 feet per second per second
it's also got a direction.
thing can accelerate that way
it can accelerate that way
if an object is moving slowly along the plus..
that's your plus axis
if it's moving that way slowly
then it speeds up
you will say
the x component of acceleration is positive
if it's moving fast and then slows down
we will ordinarily call that deceleration
but mathematically
it's negative acceleration
which means that the x component of the acceleration is negative
it could be exactly the same acceleration
as if you started from rest
and accelerated along the x axis
in both cases a change of the velocity along the x axis
is called the acceleration
and it's simply the time derivative of the velocity
all right so we write..
acceleration, the components of acceleration
are the derivative of velocity with respect to time
or you can write it as v dot of course
v dot i.
But that makes the second derivative
the second derivative with respect to time
the position
the second derivative
is usually labelled with two dots on top it
two dots means second derivative
one dot means first derivative
no dot means don't deferentiate it at all
three dots means the third time...derivative and so forth
so acceleration..
is the second time derivative of position with respect
well... second time of position period.
we can also write this in vector form
we could write
that the velocity
as a vector..
is equal to the time derivative
of..um.. sorry
time derivative of the position vector
or...
just..that gets a little annoying
r dot..
r with an arrow to indicate it's a vector
dot to indicate the time derivative
likewise the acceleration
acceleration..
is the second derivative
r double dot
OK, now that we have these concepts
let's work out an example or two.
two examples
two specific examples
of position, velocity and acceleration
First example.
Motion along a line.
Particle has a position x
this is the x axis
a particle is labelled
that's having a position x of t
along one axis
it's hardly worth thinking of it as a vector
we just think of it as x of t
it is an one dimensional vector
but an one-dimensional vectors are too trivial to even called vectors
just x of t
and that's..
as an example
let's write down a particular formula for x of t
this is x of t
um.... which I gonna assume is a constant
some number..
plust b times t
plus c times t squared
now there is nothing to prevent us from going on
But let's suppose that is the formula
which tells us where the particle is in any given time
all right?
in particular, at the start
let's take a start to be t=0
at t=0
the particle is at A
let's calculate the velocity and acceleration
you all know how to do this
to calculate the velocity
you write x dot
and that's the first time derivative
the derivative of a is..
zero, since it's a constant
the derivative of b times t?
b..
and the derivative of ct squared?
plus 2ct right?
all right, so this tells us now
that the velocity at time zero is b
to start with
but then the velocity starts increasing
of course depends on whether c is positive or negative
or whether increases one way or the other
but, whatever it is, starts at b
and then as time goes on
it either increases or decreases depending on c
linearly with time
what about the acceleration?
you just differentiate it again
what's the derivative of b?
and the derivative of 2ct?
2c
OK, so this is the acceleration
it's twice c
it's constant of time
it's constant with time
it doesn't change
so this is a uniformly accelerated particle
that's a uniformly accelerated particle
that has acceleration 2c
that's the kind of motion you have
when object falling in a gravitaion field
with a constant acceleration
but here it is
and here is the reason why
what I want to do before we finish
is one more example
which is more interesting
and it's circular motion
motion in a circle
that's I teach something interesting um.. and..no
Let's think of a particle
moving around in a circle
it start at time t=0 along the x axis
and here is y
so here is x
x and y are the components of its position
for us now
we gonna ignore the third direction
z plays no role
it's aparticle moving on a plane
it has two combo coordinates x and y
and at any given time
it's on circle
the angle increases linearly
the angle just continues to increase and increase and of course
when it goes all the way around
the angle goes from 2π back to zero
but, we don't have to say zero
we can keep..let it go one ... π and π
The angle is...here is the angle θ
and the angle increases with time
according to the rule
θ is equal to some constant called ω
times t
it's not a 'w'
its omega
the Greek letter ω
how long does it take
to go all the ways around the circle
well, to answer that
we just say: how long does it take
for it to go from θ=0 to θ=2π?
2π is 360 degrees in radians
we working in radians of course
So we simply solve the equation
2π is equal to ω times t
or 2πω
is the amount of time that it takes to go all the way around
and that's called the period
the period of motion is 2π over ω
and that of course determines what ω is
if we know the period
we solve this equation for ω
if we want a particle that moves around in a circle in the tenth of a second
we put in for the period of tenth a second
and then we calculate ω
All right, so that's what ω is
it's called the angular frequency
but...let's now consider
what are the x components
and what are.. what are the x and y components
of the position of the particle
and that's easy
oh...um...
let's have it moving around on the unit circle for simplicity
let's have it move on a unit circle
which means radius equal to 1
OK. What is x component of position?
cosine θ right?
cosine θ
what is y component?
sine θ
ok so now we know
the components of position as functions of time
x of t
is equal to cosine of ωt
y of t
is equal to sine of ωt
and now we can start calculating
the velocity and the acceleration
all right so let's calculate the velocity and the acceleration
I assume that you know how to differentiate the function like cosine ωt
so, we just have to compute
first time derivative
to calculate v sub x
and time derivative of cosine ωt is what?
minus..
ω times the sine of ωt
OK, everybody know that?
Any body not know that?
Any notes...I suspect there's a...there's a whole ... on calculus
and one other thing that I think .. discuss
is derivative of trigonometric functions
so to differentiate cosine
the direvative of cosine is minus the sine
and because of the ω here
there's a ω here
so this much I expect that you'll be able to..
you know, you recognize it
or you'll able to go home
what about v sub y?
and figure out why this is true
ω times cosine ωt
OK, now I have a interesting question for you
what's the angle
between the velocity and the position?
the position is over here
what's the angle between the position and the um...
and the velocity...
It's pretty obvious that it's 90 degrees
as you can see that just by saying
well, we know the velocity is gonna be moving along there
but, how can we check it?
We can check it
by checking the dot product..
here's.... x and y are the components of the unit vector
vx and vy are the components of velocity vector
What's the dot product between the position vector
let's call that r
and the velocity vector v?
Well it's the product of the x component of position
times the x component of velocity
plus the product of the y component of position and y component of velocity
So, it's cosine ωt times minus sine ωt
plus sine ωt times cosine ωt
they cancelled
This product here has a minus sign in front of it
this product here has no sign.
Both of them have cosine time sine in them
So, there is a direct calculation
of the dot product of position and velocity
and it's zero
what does it tell you that the dot product of two vectors is zero?
It tells you that they are perpendicular
OK. So, without...um..you know
intuitively, it's obvious that the..um..
that the...velocity
and position are perpendicular to each other
But here is the calcualtion that proves it.
OK?
Let's go on another step and calcualte the acceleration
The acceleration is another derivative
All right so what's the de... what's the derivative of minus ω sine ωt?
well, it's minus ω times the derivative of sine
which is another factor of ω and cosine
so this will give us minus
ω squared...
cosine ωt
And what about a sub y?
Derivative of ω
We first of all have a factor of two factor of ω
and the derivative of cosine?
it's equal...
minus sine
Well, let's compare a
with the position itself
The position is the vector cosine..sine
the acceleration ..
is minus ω squared times the same vector
so, apart from let's.. the ω squared is important
that's tells us how fast the velocity is
Um...
But..what important here that aprat from the ω squared
is the fact that a lies in the same direction as r itself
except not quite..
it lies in the opposite direction
there's a minus sign here
so that tells us...
that if the particle is over here..
it's acceleration...r is pointing outward
the acceleration is pointing inward
the acceleration is pointing inward toward the origin.
Well, we all know that the acceleration of a particle moving in a circle
is toward the origin
it's centrifugal acceleration
But, here's the mathematics that demonstrate it
that the velocity is perpendicular to the position of the particle
and the acceleration is parallel but in the opposite direction
How about the magnitude?
The magnitude of the position is one.
I put it on the unit circle
What about the magnitude of the velocity?
Velocity.
This is just ω right?
Just.um...you know
vx squared plus vy squared
and is equal to the square of the magnitude
sine suquared plus cosine suquared equal...add to one
So, we just left it ω squared for the square of velocity
The velocity of this particle is just ω
and that make sense
The larger the ω is
the faster the thing circles around here
and in fact...the velocity is just propotinal to ω
How about the acceleration?
The magnitude of the acceleration.
it's ω squared
OK?
So, the bigger ω is, the bigger the velocity
But even more so for the acceleration
But acceleration opposite to the direction of the direction of the position of the particle
All right, I think that's all we want to do for today
We've covered some material
I want to go through the simple elements of the acceleration, velocity,
circular motion, motion on a line
and...
and next time
we'll start to talk about what is..
what are the initial conditions for particle
What is the configuration.. what is the space of initial condition
And what are the laws of motions
what are the things which tell us how the particle..
moves from one instant of time to the next