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what we're talking about for the next
well several weeks in any case the
structure of the laws of physics period
the laws of physics all have a common
theme to them a common actually
shouldn't say an underlying basis but a
common form the laws of physics first of
all a simmer I mean by the laws of
physics there are some laws which are
mathematical laws one plus one is two I
don't regard that particularly as a law
of physics there are laws of probability
theory the fact that probability works
all together may be a law of physics I'm
not sure but I don't regatta laws of
probability theory as laws of physics
you know they're not specific laws of
physics all of the laws of physics such
as ethical czar made such as the laws of
motion of a particle in a gravitational
field the laws of electricity and
magnetism the laws of charged particles
moving in electric and magnetic fields
all these classical laws of physics oh
just a collision of billiard balls or
any such things all of those laws of
physics have a common form and the
common form is the principle of least
action the laws of thermodynamics for
example which are usually not phrased in
the form of mechanical laws involving a
principal with least action the laws of
thermodynamics are simply the
statistical laws of large numbers of
degrees of freedom given that the basic
laws of those degrees of freedom have
the form of the principle of least
action the second law of thermodynamics
is partly a law about probability and
partly a law sort of Metamora are that
whose underlying nature it is is that
the laws of physics derive from a
principle of least action so we're going
to be talking again today about the
principle of least action but before I
do as is my habit I will talk a little
just some
very elementary mathematical rules when
I said that I expected you to know some
calculus or at least at minimum to have
having remembered at least at one time
have known some Prouty calculus I also
decided that I would teach as I went
along all basic mathematics it's really
needed beyond the most elementary things
so um what we're going to need tonight
are two little theorems the first
theorem is the integrate the theorem of
integration by parts most of you
probably know about integration by parts
but since it's something we will use I
will derive it first of all suppose you
have some integral and let's make the
independent variable be T DT goes from
t1 to t2 some function but the function
itself happens to be the derivative of
some other function the F or just
ordinary derivative will do here partial
derivative DF by DT what is this
integral the integral of a derivative is
simply the function back again F but the
Internet's the indefinite integral the
indefinite integral of a derivative is
just a function itself but the definite
integral from t1 to t2 excuse me from t1
to t2 of a derivative is just the
difference of the function f at the two
end points of integration F 2 minus F at
T 1 the easiest way to see this is just
to just to think about what the integral
of a function is it's the sum of little
incremental values of the function but
the incremental values what is what is
the derivative mean
just means the difference of the
function of neighboring points so this
might be F at point two minus F at point
one divided by a little incremental dte
and then the next interval is plus F at
point the three minus F at point two
plus F at point four minus F at point
three and so forth everything cancels
out except the first and last term F two
cancels out F 3 cancels out they all
cancel out except the last one the first
one and so the integral of a derivative
is just a difference of the function at
the end points of the integration that's
number one now step number two let's
suppose the function f is actually the
product of two functions F is equal
equal to the product of little F of T
times G of T because every function can
be written as a product of some
functions but let's be specific capital
F is the product of little F of T and G
of T now what is the derivative of F
with respect to T is the sum of two
terms so let's write that as f dot
meaning derivative is equal so f dot the
left out of t times G Plus F of T on G
dot of T this is the role for taking the
derivative of a product derivative of
the first factor at the second what's
the derivative the second factor times
the first so let's apply that rule up
there F that just means the F T te and
what it tells us is that the integral
from T 1 to T
- of F dot G Plus F G dot notice the
structure of the two terms one of them
is a derivative x thing which is not
differentiated and the other is the
opposite the opposite the structurally
the second term is differentiated the
first term is not well according to the
top line that's equal to the product F G
at one end point minus F G at the other
end point let's submit the location for
this the standard notation for this
difference is just to write f of t t2 t1
as the standard notation for the
difference of a function at two
endpoints so we can write that this is
the difference of a function at two
endpoints the difference of the product
of the functions of two endpoints
oops q t1 t2 is you are
now yes
the extra via the integral of DF DT
whenever you write an integral there
always has to be a D something the FP t
is just a function I won't comment on
whether it's a stupid question but I
will say please ask me questions like
that you know when you get the little
confused about something like that
just stop me and slow me down ask me the
question and force me to make these
explicit in this one here then this is
one you're talking about right here yeah
so there should be a DTM
or let's take a special case the special
situation is with either F or G vanishes
is equal to zero at the endpoints that's
a special case which will come up time
and time again where for one reason or
another either F or G is equal to 0 at
the two endpoints it could be F being
zero at both endpoints it could be G
being zero at both endpoints or it could
be some mixed case ah but in any case if
f or g vanishes at the endpoints then
this is just zero so even that's good
enough even that's good at office but in
the special cases we'll be doing as a
rule one of these two will always be
zero and so that's not there then we can
write this formula in the form that is
usually the integration by parts formula
the integral and now I won't bother
writing the endpoints the endpoints are
implicit I'll just rewrite he went the T
to e team of F dot G now let me
transpose this over to the other side
that's equal to minus the integral same
integral from t1 to t2 dt g dot X so you
see if you happen to have an integral
with the integrand the thing you're
integrating happening to be a product of
a derivative times another function you
can interchange which one is
differentiated and which one is not
differentiated but at the expense of a
sine you have to change the sign when
you change the place with about those
again that only is true and if the
product FG well I was going to say if
the product FG is equal to zero at the
endpoints as long as the difference is
zero at the endpoints but that's a quite
point so this is one small theorem
another small theorem is that even
smaller theorem it's hardly worth
calling a theorem I'll expose the
integration by parts again when we need
it another little piece of mathematics
this one is so trivial that's hardly
worth even mentioning but we mention
anyway supposing I have a function let
me call it a of T and here's what I know
about a of T I know that when it's
integrated let's suppose it's defined
again on Niccolo from t1 to t2 a of T of
some function immutable
here's what I know about it I know about
that when it's integrated with any
function any other function f of T when
you multiply it by any function of T
this can be any function if you take any
function multiply it by a of T and
integrate it let's suppose that we know
that the answer is 0 that's a lot to
know that no matter what function you
put here if you multiply it by a and you
integrate it you get zero what does that
tell you about eight it tells you that a
is zero and the easy way to see that is
say if it's true for any function let's
check that a of T has to be 0 at some
specific point let's take this specific
point over here then all we have to do
is to take a function f of T which is
zero everywhere has except in the
vicinity of that little point then when
we multiply the function a where this
little blip let's go up the blip
function when we multiply it by the flip
function we of course get something
proportional to a at that point and so
what we'll get is something proportional
to a at the point where the blip
function is nonzero well by choosing the
blip function to be anywhere we then
prove that a of T has to be 0 everywhere
so if we have a function which when
multiplied by an arbitrary function the
integral is always zero then that
function a must be zero exactly zero
this implies
that a is zero now you can't check this
by integrating it with a specific
function if you just take some
particular function you do the integral
and find it zero that only tells you a
limited amount of information but it's
approve now you how long how long does
it take you to integrate it with every
possible function forever so this is not
something you do by here that's probably
not long enough so as a rule you don't
actually check this by going through a
family of functions you prove that a of
T times F of T always integrates to zero
ok now we're ready for the the big one
the big one is the relationship between
the principal Instax camera which is a
global statement let me say what I mean
by a global state or in person it is
trajectories or histories if you like
histories of systems the system is
described by a set of coordinates that
we can call Q 1 to 1 and the trajectory
or the history of the system is a
trajectory as a function of time fine I
take the upward here at the plan to be
horizontal I apologize for that but
there the problem is that in elementary
mathematics the independent
variable is always taken the along the
horizontal axis physics the independent
variable which is usually a time is
always taken to be the vertical axis so
we we have to tilt our heads every now
and then the season perspective okay so
that's T and a trajectory or a history
of a system is a trajectory as a
function of T of all of the cubes the
cubes being the coordinates in other
words it's a set of functions q1 of 61
of t2 of t3 of T all of the various
generalized coordinates that describe
the system that you know them as a
function of time then you know the
history we're interested in the
histories of systems the trajectories
there's two ways of going about
formulating the laws of physics one I
will call local local means you
concentrate on a location along the
trajectory and you construct a law which
tells you how to go forward at each
point in other words given a bit of
information somewhere along the
trajectory it tells you how to go a
little bit forward F equals MA is an
example it tells you the acceleration in
terms of the position so in that case
you need to know what do you need to
know in that case you need to know the
Q's and the Q dots and if you know the
Q's and the Q dots then F equals MA e
will take you to the next point the next
point in the next point and will
eventually build up the whole trajectory
for you f equals ma I call a local law
of physics welcome in the sense that it
makes reference only to a little bit of
the trajectory a differentially small
amount of the trajectory the other way
of describing the laws of physics is
through a global closed Global global
formulation which looks at the whole
trajectory and says there's some
quantity called the action which is
dependent on the entire trajectory which
is minimized for the true trajectory for
the true trajectory the women actually
describe the system the action is
minimized and for that all you have to
give again twin points how many plough
many things you have to know you have to
know two things for each coordinate the
queuing the Q dot again you need to know
two things for each Q you need to know Q
at the beginning of the trajectory and
you need to know Q at the end of the
trajectory these are two different
formulations of the laws of physics and
they are related in fact they're not
only related they are equivalent to each
other if properly formulated now how do
you see that a global law or the
principle of least action is related to
a local low on that it's not very hard
you're just focusing a little piece of
the trajectory here focusing a little
piece of in trajectory and say I if I
know the trajectory has gone through
this point and if I know that it's gone
through that point let's suppose I do
know that it went through these two
points
then the principle of least action tells
me that it must go along a trajectory
between these two points which minimizes
the action between these two points in
other words if it minimizes the entire
action it must also minimize the
action locally between two neighboring
points because if you chose a trajectory
in here which think minimize the action
between these two points then it also
wouldn't minimize the toggle action so
there is a relationship between the
global laws and the local laws in
particular if I give you the law of
physics in the form of a least action
principle we should be able to derive
from it a differential equation which
describes an analogous data equals MA
that's what I want to do now all right
it's a few steps involved and it's
probably the hardest definition of
derivation that we had up till now but
it's not very hard alright what I'm
going to do is I'm going to take let's
call Q of T Q of T stands for the actual
trajectory that the system really takes
the solution for the problem and it's
not just one Q of T its Q 1 of T dot dot
up to Q n of T where n represents the
total number of coordinates that it
takes to describe the system it was a
system of particles then n is 3 times
the number of particles so that's our
trajectory and is represented by this
red line now let's take let's change the
trajectory looping the formula a little
bit move it a little bit by taking each
Q I of T and any to it a number alpha
alpha can be any number times a specific
F sub I of to us invent the set of
functions arbitrary set of options
doesn't matter what they are let's
invent an arbitrary set of functions as
by of T again n of them and let's add
them to Q sub I of T with a coefficient
alpha we're going to change alpha but
this ever lie of T could be anything
what happens to the trajectory oh and
one other point we're going to require
that the trajectory passes through the
initial point in the final point that
means that F I of T should be zero at
the initial point of the final points
right so fi of T is to vanish at the
initial point and the final point
well I've erased it you can get the
drift
something is vanishing at the initial
point in the final point so we can write
that F IFT fi0
at t1 and t2
what does the trajectory look like if we
shift it a little bit well it'll be
shifted that's what will happen it will
be shifted at each point T it will be
shifted a little bit by an amount which
is proportional to alpha but also
proportional to the set of functions M
PI of T so it's a little bit of a shift
of the trajectory I've drawn the zone
was just a shift to the right it's not a
shift to the right it's a shift in some
more complicated set of directions
depending on the Fi of T each of the FIS
has shifted a little bit by an amount
proportional to alpha that gives us a
new trajectory I think not a fake
trajectory but a trajectory which is not
a solution of the equations let's
calculate the action of the original
true trajectory
and let's call it the action of a
trajectory a of the cues when I shift
the act when I shift the cues a little
bit a becomes a function of alpha why
because the new trajectory depends on
Alpha and in general whatever the action
is it will depend on the trajectory so
AE becomes a function of alpha as to get
detected forms of Q it becomes a
function of alpha for a given problem
with a given true trajectory and a given
set of that sub i's a is just a function
of alphabet so function about the amount
that you shifted the trajectory the
amount can be positive it can be
negative if you shift it for a positive
shift of alpha shifts the trajectory to
the right and negative shift of alpha
chips the trajectory to the left well
let's suppose we know that the action is
minimised for the true trajectories the
true trajectory corresponds to alpha
equals zero that's unshifted completely
on shifted the true trajectory is alpha
equals zero it says that a alpha is
minimized at alpha equals zero that's
what the principle of least action says
in fact it says it for every possible
collection of functions fi no matter how
you deform the trajectory if the true
trajectory is the minimum of the action
it says that any X a of alpha will be
minimized when alpha is equal to zero
that says in turn that the derivative of
a with respect to alpha is equal to zero
when alpha is equal to 0 for alpha
equals zero
it says any change in the trajectory
parameters land alpha increases the
action and so that tells us that a is
stationary at alpha equals zero now if
that's not clear stop me and I will
explain a little more this is one of the
central we're going to talk about I
thought it took about a little bit less
time but I'm going to get back around
yeah yeah right so let's now talk about
so far all I said is suppose there's a
thing called the action which is
minimized for the true trajectory now
what is this thing that we call action
it must somehow be connected to Newton's
equations because every well content is
supposed to be the same as Newton's
equations well I'll tell you what it is
right now the action is an integral
along the trajectory integral DT from t1
from starting point
t1 to the end point t2 of a thing that's
called le l stands for Lagrangian
Lagrangian was named for L'Orange the
bronze was of course a Frenchman I think
what's better pressure look all was it
cooking us oh no no no he was Armenian
that's why you little the ground
okay so it was like the polar you sure
like Napoleon well he was course again
Columbus you know nobody nobody ever
said Columbus will spray Elfa Legrand
jr. what does the Lagrangian depend on
it depends on the things that determine
the trajectory it depends on Q of T all
of it Q sub I of T let's just call it Q
of T which means all of the Q's of T but
also the velocities the position and
every time and the velocity of every
time Q dot
so here this I haven't told you what the
Lagrangian is in specific terms that
yeah alright I actually have less time
I'll just remind you it is equal the
kinetic energy minus the potential
energy the kinetic energy typically is a
function of velocities the potential
energy is usually a function of
positions so L will be equal to this but
we'll check that we will for the moment
L could be any function of position and
velocity and the action is the integral
in other words it's the sum all along
the trajectory of little incremental
actions from one point to the next
little incremental actions which depend
on the position and velocity along here
along here along here and so forth you
airball up and that's the publisher a
definition so far letting you use the
fact that it's teammates you mister
believe it to be any function of Q if
you Bob the question is how do we find
the trajectory which minimizes this
action this is a problem in the calculus
of variations it's a problem that's
called the calculus of variations why is
it called very what's being very what's
being varied is this trajectory we
change the trajectory a little bit and
when we require that the true trajectory
is the one which gives us the least
action alright so this is the action and
of course it does depend on alpha
because Q depends on now for in fact
let's calculate let's write out
over here first what's the derivative is
that's right at us Q sub I is equal now
let me put a little symbol over this a
little hat this indicates that this is
the true solution for the mathematical
problem of finding the correct
trajectory so let's call one of these
trajectories Q in the little hat I just
put the little hat on it to indicate
that it is the true trajectory the hat
indicates we solve the problem then we
add a little bit to the true trajectory
alpha f of T and that gives us something
that I'll call human without a hat on
top of it we could even think of Q as a
function of math where it is a function
of alpha Q at any instant of time is a
function of alpha because it depends on
appala now what's the derivative of Q
sub I with respect to alpha it's just an
ordinary derivative entering the right
partial derivative the derivative of Q
sub R with respect to alpha well we just
look at this this doesn't depend on
alpha this does depend on alpha so the
derivative of Q sub I with respect to
alpha is just X sub I of T
the change in queue when you change
alpha is justify of T what about the
change in the velocity
how does the velocity change when you
change alpha a little bit well the
change in the velocity the change in Q
dot if you shift output is just f dot we
can write down here Q dot is equal to
the Q hat dot plus alpha F dot I'm
assuming incidentally that alpha is just
a number alright just a number it
doesn't depend on time it's just a fixed
number fix it once and for all and then
we're going to study the answer as a
function of that number the the are the
change in Q dot when we change alpha is
just f dot all right so the change in Q
with respect to alpha is f and the
change in Q dot with respect the outfit
is just f dot let's find out now how
much the action changes how much does
the action change when you vary out for
Liu Bei let's just differentiated the a
by D alpha keep in mind what an integral
means it just means a sum of currents it
just stands for a problem along the
trajectory so integral DT none of these
things depend on alpha the thing that
depends on alpha is the Q and the Q box
right so how do we find out how much the
action changes integral that's still
there the Lagrangian will change a
little bit because the Q's in the key
box change for example the Lagrangian
changes because
changes kiss body change a little grungy
when you change Q sub I times the change
in Q sub I when you change alpha so
that's f sub I of T this is the change
in the Lagrangian due to a small change
in Q times the change in Cuba when you
change alpha but that's not all there is
to it L also depends on Q dot so we have
to add to it the change in the
Lagrangian when you keep that changes I
have to change in Cuba which is f dot
around which is f dot of T this is the
change in the action when you change
alpha a little bit yeah one sum over I
thank you
all I've used is that the change in an
object is equal to the change in each
one of its arguments times the change in
the argument sum over all from all of
the arguments arguments meaning the
sense of the independent and the
independent variables so that's that's
relaxer size this you should be able to
looks a little bit complicated but it's
actually not very complicated we just
differentiated each one of the things
inside L and multiplied by the amount of
acting changes okay now here we have a
term that has an F of T in it remember f
of T could be any function at all with
one restriction that advances at the
endpoints we're not allowed to move the
endpoints the endpoints are nailed in
place but here we have F dot I really
would much prefer to have F there then I
will have both of these terms
multiplying an arbitrary function what's
the trick preparing this and - f
integration by parts you can take the
one which is differentiated it which is
differentiated and undifferentiated if
you take the one which is
undifferentiated and differentiate it
alright but you also have to change the
sign so let's just work on this I won't
bother rewriting it let's just work on
it we're going to undifferentiated EV at
the cost of differentiating this whole
thing here I think I'd better I think
I'd be better off to rewrite it yes I'm
going good what is G first or what is G
and what is F at this is that but G is
the L by T Q dot I'm not fiddling around
with this term of only fiddling with
this term here and for this term the L
by T Q dot plays the role of G we have F
dot times G okay so it's put a red
circle around the thing that I'm
identifying is G and a red circle around
G over here okay so what do I get then
for the second term the second term
along the second term becomes minus
integral DT from t1 to t2 DT and now F
times G dot that means that means D by
DT D by DT of the derivative of L with
respect to Q bottom and a complicated
looking thing D by DT
of the derivative of the Lagrangian with
respect to Q sub I dot times X sub I
derivative it's here this is G dot D by
DT of G and instead of having a thought
we now have that so now we can write
both of these terms both the terms the
term coming from varying q a little bit
and the term coming from very acute out
a little bit as integrals with the
arbitrary function f let's write the
whole thing down now this one came in
with a minus sign this term and the
first term was just the L by DQ
I've survived
in a row
Petey
whole thing gets integrated here again
I'll just remind you where this this is
a term the action changed because we
changed cue a little bit this is the
change in the action because we changed
cue top a little bit
what about the what about the
contribution from the endpoint so the
integral remember there was another term
which was F times G at the endpoints why
do they match up they're stipulated that
you want those guys we stipulated that
the trajectory passed through the
initial and final point so that said
that F should be zero at the endpoints
so there's no contribution here well
what like what are we calculate we've
calculated the change in the action with
respect to alpha change in the action
with respect to alpha and alpha equals
zero we can now set alpha equal to zero
doesn't matter doesn't even depend on
alpha what's the rule the rule is that
this should be equal to zero
that's the condition of minimizing
action with respect to alpha so this
should be equal to zero notice that the
X can be taken outside the bracket oh
probably aware that there should be a
sum of an idea each term in the sum is f
sub I multiplies f survive
this has to be zero for every possible
function f survived all possible ways of
deforming the trajectory why because
we're looking for the trajectory with
the minimum action and so however we
deform the trajectory we should increase
the action a little bit so it follows
then that this integral has to be zero
for every possible function or what does
that imply that implies that the thing
in the curly bracket here has to be
identically zero so what we've done is
we've taken the condition of lease
action and we have converted it to a
local state rate at each point along the
trajectory at each point along the
trajectory we do incidentally this is
true for every combination of functions
at 1 of 2 or 3 and so forth that means
that every one of these expressions has
to be equal to zero
but each body at every point in time the
L by DQ minus D by DT of the L by DQ dot
has to be zero all along the trajectory
this has the form of a local equation
and each point on the trajectory it's
equivalent to the global statement to
the action is minimum okay now the way
this is incidentally this is called the
Euler Lagrange equation for the
variational problem of minimizing the
integral of L this is called the Euler
Lagrange equation or Euler not Oh
like the football team project Euler
Euler Lagrange variational equation and
this equation is the heart of all of
classical physics in fact in a slightly
different form they related but the most
sophisticated form it is also the heart
of quantum mechanics the form of a
hundred packet the growth every physics
problem known to me or every physical
system known to me probably known to
anybody can be formulated as an action
principle with the Lagrangian that
doesn't only include mechanical systems
and include systems of fields
gravitational fields general relativity
string theory everything as a form of a
variational principle you need to say a
principle of least action let's say in
English what is what say this final
statement
well we're going to the best way to
let's come back to what it says in
English after we work out what it says
in specific examples perhaps by the time
we say so put on specific examples we'll
understand what it says what does it say
let's say F equals our aim is that is
that English enough so that's the next
thing to do is to show that in a couple
of cases of interest
it really does produce the right
equation of motion all right but in
English really what it is is it's the
equivalence between a global description
of the trajectory and a local
description alright the local
description is a differential equation
it's a differential equation as we'll
see we'll work out some examples or it's
a system of differential equations of
the form F equals MA incidentally just
in terms of those some some lingo some
definitions young birth in equating that
to
yeah yeah you're asking is there another
way to say is that yeah yeah the time
derivative of the derivative Lagrangian
with respect to a velocity let's call it
is equal to the derivative of Lagrangian
with respect to the corresponding
coordinate all right but I mean the best
thing to do let's write it let's write
it up here and then work it out to some
examples to see how it works
velocity something at the pass Pacific
point yeah right all right let me give
you some names the quantity DL by D Q
sub I dot notice that it depends on eyes
or depends on which coordinate we're
talking about that is in need it's
called first of all it's got a symbol
associated with it it's usually called
PI sub I it's the derivative of the
Lagrangian with respect to the velocity
gives you survived dot so it's a
component of something that's related to
the derivative of the Lagrangian with
respect to a component of velocity if
you like you may know the name of this
is it's the momentum canonically
conjugate to Q the Q sub I or you can
just say the momentum the momentum
component which corresponds to the
position component gives the body if you
think of the Q's as a bunch of positions
then the paisa bodies are called the
momenta now let's all right so what does
it say this is D by DT of something that
I'm calling a momentum is equal to DL by
D Q how about this this kind of name the
generalized force so this is this is the
I'll just call it the momentum but the
planet is called the canonical momentum
conjugate or the momentum conjugate to Q
a moment
conjugate the queue for every queue do I
pick you I thank you on the other hand
the L by D QR e is called the
generalized force or generalized
component of the force
I won't bother writing component you get
the idea so what this says is the
derivative of some momentum is equal to
some force doesn't get people's M&A in
these articles are many let's try it out
for a specific Lagrangian t minus u
let's first do it for a single particle
1 particle mass n for the moment one
direction of space just as simple as
possible problem then T is equal to 1/2
the mass of the particle times X sub I
dot squared just x squared it's a 1 1
coordinate simplest possibility that's T
the kinetic energy X times of course
just velocity and describe the skies
minus U of X that's L what is the cue
for this problem I'm not using the
language of Q's at the moment killers
are plus tax
so I'm reserving X now for linear
position along the line but of course it
is the cue for this problem here is just
a generalized term for any kind of
coordinate X is usually used when we're
talking about rectilinear coordinates or
Cartesian coordinates alright let's work
out what this what the Lagrange's
equation said what the Euler Lagrange
equation say first of all what is the L
by D now it becomes the L by the X dot
this is a canonical momentum conjugate
to X well the L by DX dot is X dot the
only place where X that appears is just
equal to MX dot the 2 goes away because
we have X dot squared and when we
differentiate with respect to X dot
let's just give this in next time that
is what we usually call the momentum
sometimes called pee okay so here's a
language when we're talking about
ordinary coordinates along lines X Y Z
we use the language X but momenta we use
the language P when we're talking about
more generalized kind of coordinates we
use the language Q and PI but in this
case Q is X and pi is P what does
Lagrange's equation say it's is D by DT
of DL by DX I'd but that's MX dot is
equal now let's equals the L by DX the L
by DX is minus the U by DX the only
place where X appears is in you X dot
appears here X appears here so this is
just minus D u by D X this should be
familiar this is just that pickles n a
mass times X double dot is equal to
minus 2 u by D X so we successfully
reproduce Newton's equation for a
particle with one degree of freedom
moving along a line with a potential
energy you notice that had I had I used
for Lagrangian plus here I would have
gotten the wrong sign in the force law
here so the energy which we haven't
talked about yet tonight very much other
but the energy which is t plus u is not
the laganja the Lagrangian is t minus u
and Lagrangian is not conserved
Lagrangian is not typically not a
conserved as the particle moves along
the orbit in general Lagrangian will
change its T plus U which doesn't change
but we have to come back to that we
wanted to write want to derive energy
conservation from this rule here but so
far we have we do have the run through
situations okay now let's go forward a
little bit to a mini particle system
from any particle system only have to do
is for each particle let's have many
particles and many coordinates for each
particle three coordinates for each
particle XY and Z let's just call them
x1 x2 and x3 but also we have all of the
particles
let's lump them all together and call
them x1 dot dot dot dot down to X n
where n is three times a number of
particles 3 9 4 3 and let's just use to
be capital m for the total number of
exits now each particle might have a
different mass so that means each X dot
squared might have its own mass that
goes along with it
of course they'll come in triplets are
the mass of a particle doesn't depend on
which way it's moving we can think a
little more generally we can
imagine a slightly more general
situation where every X dot squared goes
with its own mass this would be a
mathematical for real physical particle
of course the mass doesn't depend on
which way it's moving so it triples they
would come together with the same mass
that's all right
we can we can extend the notation a
little bit and just write that the
kinetic energy is the sum over all the
coordinates of the appropriate mass for
that coordinate times X sub I squared
divided by 2 now this doesn't
I didn't just mean one two and three
meaning XY and Z and it doesn't just
mean which particle we're talking about
it means the collection of all the
different things which specify a
particular coordinate which particle as
well as which direction and then the
potential energy which is a function of
where the particles are it's a function
of the positions of the particles just
becomes minus some potential which
depends on all of the particles I think
that the change is minus u of X U of X
here means you will have all of the
coordinates now in specific instances
you may have some particular special
form and we'll talk about the special
forms that it can have but in a general
context we can imagine that you which
just depends on all of the positions of
all particles then what happens if we go
through the same exercise coordinate by
coordinate well the L by the X sub I God
becomes n sub I times X sub I that
we'll call that piece of I this does
know something ended here Oh summation
entangled yeah so if this is X Y or Z
this could be the X Y or Z component of
a particular particle momentum but we
could also vary which particle we're
talking about
and so this extends not only over the
directions of space for which particle
we're talking about there are three end
momenta and three end positions each
equation then takes the form of the time
derivative of the momentum of a
particular direction and a particular
particle is equal to the force the
component of the force associated with a
particular particle and the particular
direction of space this again is F
equals MA F equals are made for each
particle in each direction of space so
you Brady X survive
the implication is that this must be
very very general after all this is is a
collection of particles with force laws
the force law is happening to be just
such to hold it together become an
eraser you're nothing but a collection
of particles held together by forces
this is an extremely general system that
we're talking about here and it is
definitely governed by a principle of
least action the principle of least
action being gotten by taking few - by
taking kinetic energy minus potential
energy so we see already that the
principle of least actually is general
enough to cover just about every
possible mechanical system we could
think about because every mechanical
system we could think about we could
imagine being made out of chunks of
material which in turn are made of our
collections of particles held together
by inter atomic forces or whatever holds
their the system together let's do a
couple of examples just to see how
things work
let's do an example of two particles on
the line interacting with a force
between them governed by a potential
energy all right so we have two
particles on the line particle one has
coordinate x one particle to has
coordinate X two but if one has mass and
one particle two has mass and two now
there's also a potential energy function
the potential energy is a function of
both coordinates if there's a force
between the two particles and the
particles influence each other then you
would expect in general that the
potential energy will depend on the
position of both particles maybe it's a
complicated way now I want to take a put
them on a focus on a particular
situation and the particular situation
is one which has what I'll call for
everybody calls translation invariance
translation invariance means that for
purposes now the Lagrangian or let's say
in particular the potential energy would
not change if you move both particles
together in other words it doesn't
depend on where the particles are it
does depend on the distance between them
this assumption is equivalent to any
symmetry the symmetry being explained
what I mean by symmetry in a moment more
generally but the symmetry being
translation symmetry nothing depends on
where you put the origin of coordinates
things only depend on the distance
between particles which is another way
of saying that if you move the whole
sister if you transfer
later translation means moving it
rigidly everything being kept at the
same distance that in this case the
Lagrangian would not change the
velocities don't change if you just take
a particle moving with a given velocity
given velocity moves means it moves a
certain distance in a certain time if
you change both the initial position and
the final position by the same amount
then the velocity doesn't change in
other words if if your car goes one mile
in 10 seconds spots fast now but if you
move it ahead by a certain distance and
you keep the velocity the same then it's
still going one mile you know what I
mean velocity itself doesn't depend on
the choice over coordinates so the
choice of origin why well let's think of
X of T and let's say that a constant to
X of T just add a constant to it if you
add a constant to it it doesn't change
the time derivative rx so shifting
everything by a constant amount would
change none of the velocities and if the
potential energy only depends on the
distance between them then the potential
energy doesn't change if you originally
move everything under those
circumstances the Lagrangian
or the system has a symmetry it doesn't
change if you if you move everything by
the same amount so specifically then in
that case the Lagrangian for this two
particle system would be kinetic energy
m1 x1 dot squared divided by 2 plus n2
x2 dot squared divided by 2 those are
the kinetic energies and then the
potential energy
- and the potential energy now is
assumed to only depend on the distance
between x1 and x2 so it only depends on
X 1 minus X 2 now this is not be the
case if there was a large lump of matter
out here that happened to be exerting
forces on both X 1 and X 2 then the
potential might also depend on how close
you were to that lump of matter we
wouldn't just depend on the distance
between them it would also depend on the
distance of a big lump over here but if
the only system happened that the system
is a closed system containing only 2
parts and which translation invariant
meaning to say everything everything
stays the same if you shift everything
then the potential energy can only be a
function of the distance between the two
points x1 minus x2 actually it doesn't
have to be a tab solute value but we
could make it absolute value if we
market the runner to be literally the
distance between the two points then
then we make an absolute value but for
my purposes right now it's good enough
that it's just a function of the
separation the x1 minus x2 that's good
enough that's the translation invariant
that means that if we were to add a
constant that each X the same constant
to each X nothing would change the
velocities wouldn't change because the
velocity doesn't depend on City + x1
minus x2 with change because the
constant translation would cancel out
all right so this is a symmetry this is
the case of translation symmetry
let's workout now Lagrange's equations
is an equation for x1 is an equation for
x 2 it the equation for x 1 is d by DT D
by DT of partial of L with respect to X
1 dot equals partial of L with respect
to X 1 alright d by DT of partial of M
with respect to X 1 dot that's just M a
for the first particle right that's just
that's just D by DT of P 1 P 1 by DT D 1
being the momentum of particle 1 the L
by D X 1 dot is just M X 1 dot that's
the momentum and so the first the
equation for the first particle starts
with the P 1 by DT is equal to DL by DX
1 alright only thing that depends on X 1
is U so that becomes is equal to minus D
u by DX 1
what about the second particle that's
particle number one let's do particle
number two now VP 2 by DT let's just
close from here that's going to equal
minus vu by the X 2 what's the relation
between the u by DX form in the U by DX
2 they're equal and opposite if a thing
only depends on the difference of two
variables like this then the derivative
well let's just look at it the
derivative of U with respect to X 1 is
just a derivative of U with respect to
the argument is it not is it obvious to
everybody that the derivative with
respect to X 2 is minus the derivative
with respect to X 1 if it's obvious to
everyone I won't explain it it's okay
all right so let's let's do with that
whatever function which only depends on
X 1 minus X cube now let's just call
let's just call this variable X 1 minus
X 2 let's give it a name let's call it X
so U is only a function of X now let's
not call it X let's call it D for
difference D for difference so U as a
function of D D is x1 minus x2
all right let's calculate first of all
the derivative of U with respect to x1
by the chain rule the derivative you
will respect x1 is the derivative of U
with respect to de remember U is really
only a function of B times the
derivative of the
with respect to x1 what's the derivative
of B this is the XOR minus X to be 1
minus x2 what's the derivative of B with
respect to X 1 1 1 1 so this is just be
you by DB what's the derivative of U
with respect to X 2 that's the U by D D
times the change in D when you change X
to once the change in U when you change
X to what's the change in D when you
change X to minus 1 the derivative of B
with respect to X 2 is minus 1 so the
minus sign between them
derivative with respect to one variable
is minus the derivative with respect to
the other variable that's as a
consequence of the fact that it only
depends on the difference between them
well the implication seems to be
Newton's law of action and reaction
limit is the fact or the assumption that
the potential energy only depends on the
distance between network difference
between the two variables that it's
translation invariant there's a
translation symmetry that's equivalent
to the statement that the force on
particle one is minus the force on
particle two what's the net result the
net result is momentum conservation if
you add these two equations to find the
P total P 1 plus P 2 by DT then the
forces on the right-hand side will
cancel just as they do just this Newton
assuming that they work so what we find
from this is that momentum conservation
for this simple system here is a
consequence of translation invariance
it's a consequence of translation
invariance number one and number two the
form of the equations and the fact that
they derive from from the principle of
the least action well we're going to
explore this more thoroughly so far this
is just an example and it's an example
of the relationship this is really
really important this is maybe the most
important thing that you can learn
tonight or that you can learn about
theoretical physics practically is the
connection between symmetry and
conservation laws here's the first
example of the relation between the
symmetry translation and variance in
space and a conservation law which is
the conservation of momentum the
symmetry says that you only depends on X
1 minus X 2 for that it follows that the
force on particle
one is minus the force on particle two
and from that follows conservation of
momentum conservation of linear momentum
will call it so this is the first
example and we'll see more of them of
conservation laws and their
relationships to symmetries as I said
this is a very very deep underlying
theme of physics of theoretical physics
that every symmetry has associated with
it a conservation law what is the
symmetry mean the symmetry means an
active operation that you can do on the
system for example translating it which
does not change the Lagrangian which
does not change the value of the
Lagrangian or better yet the value of
the action if the action of a system
does not change when you do a
transformation of some type in this case
there's moving everything to the right
by a fixed amount R that is a symmetry
of the particular action that we're
talking about every time the action has
a symmetry there is associated with it a
conservation law this is the simplest
example of that and the first it goes
first of many examples that I hope we'll
see over the next however long we pursue
this question
ah yes yes very good in particular there
would be an inversion symmetry if this
were a function only of the absolute
value yeah okay in classical mechanics
that's called a discrete symmetry a
continuous symmetry is one that you
could imagine doing a little bit or a
lot we'll talk about this difference
between discrete symmetries and
continuous sentence but a discrete
symmetry is one that depends on a
parameter that can be very small just a
very small shift or very big or that you
could build up the big shifts out of a
lot of little small shifts that's called
a this a a continuous symmetry our
discrete symmetry is an operation like
interchanging two things which only has
a discrete form since in one way or the
other and there's nothing interpolating
between them in classical mechanics
discrete symmetries don't generate
interesting conservation laws and
quantum mechanics they do so but that's
that's for a later later discussion so
you're right it does have another
symmetry in particular if we say this is
a function of absolute of X 1 minus X 2
this wouldn't have much on both but
implications in classical mechanics if
those have implication in auto mechanics
implication of a conservation law
ok um so we just did conservation of
momentum for simple K let's do a couple
of more mechanical cases of the more
mechanical systems to get the feel for
this let's do a problem in which a
particle moves in two dimensions
particle moving near the surface of the
earth is the surface of the earth is our
particle a turret yeah of course it has
three coordinates associated with it but
so just pick two for its X&Y X goes this
way why those vertically upward the
particle is described by two coordinates
there are two Q's now not because there
are two particles but because there are
two directions of space x and y the
kinetic energy is one-half M X dot
squared plus one-half and Y dot squared
I have assumed that the mass does not
depend on which direction the particle
is moving we could have imagined some
sort of nutty particle who happen to
have a larger mass with respect to
vertical motion than horizontal motion
but about the right now and then what
about the potential energy all right
this is the gravitational potential
energy that I'm talking about the
gravitation in case I haven't said it
before I will tell you now the
gravitational potential energy of an
object near the surface of the earth
this is not in a situation where we're
far from the surface of the earth but
near the surface of the earth is just
proportional to the height to the
elevation so you subtract the potential
energy the potential energy is equal to
the height of the particle why it's
proportional to its mass the heavy of
the particle and more potential energy
it has when you lift it and it's
proportional to the acceleration of
gravity G 10 meters per second per
second Fuji is eight and metric units
and it's just the strength of the
Earth's gravitational field near the
surface of the earth okay let's see
let's see let's as what is a force on a
particle the force on the particle has
no X component because the u by DX is
zero the you by dy is equal to mg but
the force is - fat so - the UID y is
minus mg and the minus sign represents
the fact that the force is downward when
we take the y axis to be pointing upward
towards accelerated boundaries all right
so let's add now or subtract as the case
that be the potential energy - ng y
what they have symmetry does this
problem have well it has translation
symmetry along the x axis if I change X
by a constant doesn't change X dot and
since the potential energy doesn't
depend on X from our on Gian
this is laranjinha Lagrangian is
translation invariant along the x axis
it has X translation how about Y
translation invariance let's we'll have
Y translation invariance no why because
of the Y over here the USB Y for a small
Y for small Y why life is more wine it
was on first punch who's the president
of China
so small wine this moment Morita when
you change why the lagrangian changes
alright because of the why this here now
notice incidentally let me point
something else out we could add a
constant this is something I should have
pointed out a while ago you can always
add a constant to the potential energy
at no cost and changes none of the
physics now that constant has to be
treated as a true constant throughout
the problem it doesn't change anything
because it doesn't change the force do
you buy it well the you buy if you add a
constant that doesn't change anything
written of you and so it doesn't change
any of the forces it resets the zero of
potential energy but there's no physical
consequence of adding a constant the
potential energy so it doesn't matter
where you measure the altitude from
whether sea level or whether it's ten
meters above sea level and so forth but
if you change Y you change you all right
so that logic we should find one
conservation law not to we should find
one conservation law associated with X
and not the corresponding conservation
law with respect to Y because in handy
one symmetry and in fact if we work out
the equations the equations of motion
will be the L by DX times and X dot D by
DT of that that's called ma mass times
acceleration and that's equal to zero Y
zero because you doesn't depend on X
this is just conservation of the X
component of momentum so the X component
of momentum is conserved but the y
component
D by DT of my dot that's a time
derivative of the Y component of
momentum is the derivative of U with
respect to Y derivative of minus u with
respect to Y so that's equal to minus mg
minus mg the u by dy is minus mg or
minus 2 u by dy at minus mg all right so
what is this equation this just tells me
that the vertical acceleration is
proportional to the force of gravity or
or that the acceleration is equal to the
acceleration of gravity cancel out the
MU cm but notice important thing is that
there is no conservation of wine
momentum of the Y component of momentum
and that's associated with the fact that
there is no symmetry associated with
translation in the Y direction I'm going
to do one more example of a relationship
in particular the relationship between
symmetry and conservation laws but also
it's an example in which I won't use
rectangular coordinates that's across
this classical mechanics brought her
falling body when he remember right this
founded westward I'm saying you would
say that there is conservation of energy
potential energy and the kinetic energy
remain constant so it's the reason yeah
so far so far I haven't even talked
about in this context conservation of
energy yeah we will talk about it and
you'd be perfectly right we'll find but
in this structure there's a conservation
not of kinetic energy minus potential
energy but kinetic energy plus potential
energy so it's going to require an
action step to see that the energy is
conserved and we're going to want to ask
the question if energy is conservative
is it also conserved because of some
symmetry and if so what symmetry so we
want to we want to formulate our laws of
the basic law of physics now is that
symmetries equal conservation laws and
we're going to want to see that mat that
follows from the form of the branches
equations but slow luffy Kalulu bill
steps a little feet the principle of
least action does not really depend on
the coordinates that we use to describe
the particle whatever coordinates you
use you can use polar coordinates you
could use rectangular coordinates you
could use parabolic coordinates you
could use some screwball coordinates
with curve the fancy curve the axes and
so forth you can always
normal coordinates which with curved
axes and so forth use any coordinates
you like you can rewrite the expression
for the kinetic energy you can rewrite
the expression for the potential energy
it may look more complicated in another
coordinates but you can rewrite the
expression for kinetic energy potential
energy and Lagrangian and you can
rewrite the expression for the action
the action may look more complicated in
terms of other coordinates but it will
still be the same good old action it
will depend on the trajectory and give
you the same answer that you would have
got the rectangular coordinates if you
rewrite it correctly the fact that the
action is minimized for the correct
trajectory is a statement that doesn't
depend on the coordinates use so the
principle of least action is a principle
that is coordinate independent it
doesn't depend on the specific
coordinates and therefore you can apply
it to any set of coordinates you use to
describe a system I'm going to show you
how you would do it with polar
coordinates if we were talking about a
particle again let's take the case of a
particle moving in a plane and now
instead of describing it by X&Y
let's describe it by r and theta
distance from the origin are an angle
relative to the horizontal theta we have
to write the Lagrangian then we'll write
Lagrange's equations the system with two
coordinates R and theta and a trajectory
will now consist of an R of T and a
theta of T in other words at every
instant of time the distance from the
origin and the angle that determines the
trajectory of particle okay let's work
out the kinetic energy first the kinetic
energy is one-half M
squared so we have to do is we have to
calculate the magnitude of the velocity
to do that let's temporarily set up some
coordinates at this point over here at
the point of the particle where one axis
is along R and the other axis points in
what I'll call the theta direction sort
of on the angular direction but
temporarily for the moment let's just
think of them as Cartesian coordinates
except erected right at this point then
the velocity has two components one of
them is the component along R and you
will not be surprised that will not work
this out for you but it won't be
surprised I think if I tell you that the
component of velocity along R I'll write
as V sub R velocity along the r
direction is just r dot r is changing
then there's a component of velocity
alone r it's just the r Bob what about
the component of velocity in the
perpendicular direction
how big is if you might think it's just
data dot this is V along the other axis
I don't want to call it B theta I'm just
going to call it V along the
perpendicular axis perpendicular to R
how big is that
it's not theta-dot but it's our theta
god how do you see that well supposing
your particle is near the origin over
here and you change theta a little bit
per unit time then the mozo element has
a certain velocity suppose you change
the angle by the same amount but the
distance of the particle is far away
then it moves a lot more so the further
out you are for a given theta dot for a
given theta dot the further out you are
the faster you're moving right the tips
of my fingers is moving faster than my
elbow and in fact the velocity is
proportional to how far away you are but
it's also proportional to theta dot
because if they thought was zero if
there was no change in the angle you
would have no component of velocity
along that direction so these are the
two components of velocity and the
kinetic energy is one-half M times the R
component of the velocity squared R dot
squared plus one-half M times V
perpendicular squared which is R squared
theta dot squared this is the kinetic
energy
now let's invent the potential energy
I'm going to invent a special kind of
potential energy which is called the
central force central force means that
the potential energy only depends on
distance
sorry are not squared thank you I got
squared n V squared 1/2 mr dot squared
and 1/2 M R squared times theta dot
squared now let's introduce a potential
energy and let's make the potential
energy that only depend on R only depend
on distance that's a special case but
it's a special case with symmetry what's
the symmetry that's involved if I say
that the potential energy only depends
on distance rotational symmetry what
that would say is that nothing changes
if I rotate everything if the potential
energy only depends on distance only
depends on distance that means the
potential energy does not change if I
rotate the position of a particle or if
I rotate the axes if I just physically
take the particle and rotate it without
stretching R then the potential energy
doesn't change if the potential energy
only depends on R
yes
there's depend on finger yeah yeah but
it doesn't it doesn't it kinds only
unfaded bad so we change all Thetas by
the same amount
yeah that's correct and we have to check
that but as you can see because it only
depends on theta dot and not theta
explicitly then if I shift all Thetas so
they say theta you're lining so then we
have minus U of R minus U of R let's
work out now the Lagrangian of motion
the Q's of R and theta
so all we have to work out is first of
all for our or start with our partial of
L with respect to r dot incidentally
this is our L is kinetic minus potential
partial of L with respect to R naught is
mr dot that's also equal to the momentum
conjugate to our it's just a radial
component of momentum in our dot or M
times the velocity of your direction not
surprisingly that's the radial component
of the linear momentum then your grandes
equations become D by DT of this which
is d by dt of mr dot also known as mr
double dot the radial component of
acceleration is equal to minus d u by dr
obviously - to you by the R is the
radial component of force this is just a
radial component of s equals MA the mass
times the second derivative of R is the
U by dr notice that the radial component
of the momentum is not conserved
we can write this
as D by DT of the radial component of
momentum not equal to zero but is equal
to minus D u by D R so there is no
conservation of radial momentum let's
look at the angular equation of motion
ground the angular equation of motion
equation of motion for theta same deal
this think of theta as a generalized
coordinate partial of L with respect to
theta dot that's in R squared theta dot
what's then what's its name
what's the name for that object the
canonical momentum conjugate to theta
what's it gonna mean for angular
momentum all it is is angular momentum
or we see now angular momentum arising
out of the same pattern DL is DL by D
theta dot same time derivative of L with
respect to Q da is the momentum
conjugate to Q the L by dr dot momentum
conjugate to R the L by D theta dot or
make them conjugate to theta now now
what is the equation of motion for theta
it's D by DT of this D by DT of M R
squared theta dot is equal to 1 0 in
this case because you doesn't depend on
theta by assumption
D by DT of something equals zero that
says that that something doesn't change
so this is an equation of conservation
it's conservation of M R squared theta
dot or conservation of angular momentum
when the force or when the potential
energy doesn't depend on angle then then
angular momentum is conserved it's
usually called L incidentally this is
usually called L so it says that L is
conserved
in fact that allows us the vector that
gives us a good clue about the nature of
the motion of the system if fellows
consider let's just see what that says
it says that M R squared theta dot
doesn't change so it says that in R
squared theta dot is equal to a constant
it doesn't change with time it's
whatever it was when you began however
you begin to motion the angular momentum
doesn't change well you can solve this
and write that theta dot is equal to the
constant of M R squared so you see what
happens as R goes in and out for
whatever reason as are perhaps the
oscillate sort gets bigger or gets
smaller theta dot changes to compensate
this is just the the spinning ice skater
effect as R gets smaller plate that gets
bigger it's the conservation of angular
momentum but you see it arises once
again from a symmetry the symmetry in
this case being rotational variables
rotational invariance angular momentum
conservation translation invariance
linear momentum linear conservation of
momentum that's the pattern
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