We have seen, so far a few general properties
of dynamical systems. I would like to backtrack a little bit and go back to a formalism called
the Lagrangian formalism, for dynamical systems. Specifically, created in order to reproduce
the equations of motion that we know of the systems we already have, familiar with and
to give new incite into other kinds of dynamical systems.
This is a very general formalism and it kind of explains why Newton's equations arise?
So, you might have wondered from your elementary courses on physics, why do we have Newton's
equation of motion? Is there something deeper some underlined structure from which you get
Newton's equation of motion. And the Lagrangian formalism answers this question. So, let me
go back and talk about mechanical systems for a while, then we come back and try to
put it in the general framework of dynamical systems. In particular, I would like to introduce
the Hamiltonian formalism which is going to play important role for us in statistical
mechanics and quantum mechanics. So, as a backdrop let us ask, where do Newton's
equations come from? For instance the force is equal to the mass times acceleration for
a single particle, moving in space, where does this come from? Was it just written down
by Newton and we require, it was based on experience, based on experimental evidence,
but is there a deeper principle. It turns out all equations of motion in nature, both
classical and quantum mechanical can actually be obtained from certain principles fundamental
principles called action principles. And the extremization of the action which, I am going
to define leads to the equivalent of Newton's equations of motion, for the simple systems
we have used. Let me give an analogy, if you have a particle,
system in static equilibrium then, we know from elementary physics, that the position
of static equilibrium is given by the minimum of the potential energy; system tends to minimize
its potential energy. That is an example of an extremal principle, something is minimized
is that an extreme. Similarly, if you have in thermodynamics, if you have a isolated
system thermal equilibrium, then we know the entropy is maximized. And again you have a
extreme points, if you have a same similar thermodynamic system, at constant volume and
at constant temperature; then the Helmoltz free energy is minimized in a state of thermal
equilibrium. If this were at a state of constant pressure
and constant temperature, then the Gibb's free energy is minimized as a function as
a function of variables on which it depends such as pressure and temperature. So, in all
cases we have an extremal principle and the interesting thing is, we discovered that even
in dynamics even when things are moving there is such a minimization or extremization principle
in operation. And I am going to introduce this without much further do state what the
lagrangian is and when we will come back and show that it yields the equation of motion
in the cases we want; and goes further and helps you to find the equations of motion
in more general cases. So, this is going to be the Lagrangian formalism,
and we have in mind a system described by certain general coordinates q q 1, q 2 etcetera
q n; and generalized velocities q 1 dot, q 2 dot, q n dot, and possibly the time itself,
you have a function L which I call the Lagrangian, which I will write down as we go along.
This function L is supposed to be given to you, for a given system; you are supposed
to discover what the independent generalized coordinates are, corresponding generalized
velocities and then write down the Lagrangian and obtain the equations of motion for the
system from the Lagrangian. This is the program and the important question
of course, is where does this Lagrangian come from, where do you get it from. We are going
to a backwards guess the Lagrangian and the cases for which, we know the equations of
motion and then generalize from there. There are a general principles which helps you to
write this Lagrangian down and this is a very cumbersome thing to write down. So, let me
just write this as L q q dot for short. But always I have in mind the system with
an arbitrarily large number of degrees of freedom 1 to n. We will not look in this course
at systems which have an infinite number of degrees of freedom; we will come to that when
we do statistical mechanics. This n is some finite number and then the Lagrangian is functions
of q's q dots which are independent dynamic variables; and possibly if the forces on the
system are time dependent, if it is non autonomous, there could be an explicit dependence on t
as well. The statement is when, that if an free space for an instance in q q dot space
you had some point here at time t 1, the system is at this point in free space, that is some
initial time q one arbitrary initial time t 1 and it reaches this point at time t 2.
It moves along a phase trajectory clearly, this could be the phase trajectories systematically
along which it moves in this multidimensional free space. And the statement is of all the
possible motions from t 1 and between t 1 and t 2, the system chooses that particular
part, along which this quantity t 1 to t 2 L d t call the action; this is defined as
the action and of course, it is a function of t 1 and t 2. It chooses that path for which
the action is at the extreme, not necessarily a minimum in all cases could be a maximum
also in certain cases. But we will call this the principle of least
action, because in most of the cases in which you are really interested it is going to be
a minimum. So, it could have chosen this path for instance or it could have chosen that
path; but it chooses that path that trajectory phase trajectory along which the action is
at a minimum, is extremized which means that the path is determined by saying delta of
t 1 t 2 L dt is equal to 0 where this stands for the variation of this quantity.
For instance, if this is the q direction and between this point and this point this is
d cube delta cube variation. So, all possible variations, that we take into account and
then of all those variations the system will choose the path along with the variation of
these action between p 1 and p 2 is at an extreme. This is the principle of least action,
I am not going to try to justify here and to prove it, but this is what experience tells
us that, this is what leads to the equations of motion. It is very satisfying because,
just as in all static situations you have a certain minimizations something is minimized;
the potential energy is minimized, the minimum of the potential energy. Similarly, you have
a minimum of something else even when you are moving, and it is the action.
Now, this leads to equations of motion, because it is really a statement which tells you how
the system is going to propagate from this line to this line along this path. Therefore,
you could expect that even though this thing here looks like variation of some integral
between t 1 and t 2 some integral quantity is at 0, the first order variation is 0 even
then it will turn out that since t 1 and t 2 are arbitrary. At every point in the path
you can apply this principle and you should expect you will end up getting a differential
equation for the motion. Because I could choose this point to be t
1 and that point to be t 2 and its still true between those two points. So, the action is
actually extremized all the time and therefore, we are going to end up with a differential
equation and how do you do that? Well, the algebra is quite simple, I will write this
down and I am going to keep t 1 fixed t 2 fixed for the moment just to get the equations
of motion. And once you are here or here, definitely
there is no delta q here, there is no variation here everything all the parts join up here.
So, you are looking at all parts you start here and which end there. So, there is no
delta q at the two end points. Therefore this principle reduces to saying
t 1 to t 2, 0 is equal to this and times delta L, inside all possible shifts in L induced
by a shift in the q. So, if you arbitrarily set instead of q, q
plus delta q, L changes something and all such delta L possibilities and then you equate
as 0, but this is easily written down as t 1 to t 2. L depends on these variables therefore,
this is delta L over delta q by the way I have n degrees of freedom, but just for simplicity
of notation. I am going to write this thing as q, so it is understood that there is a
q i here and all the i's are independent vary independently I will call them collectively
this. This stands for summation over i delta L over delta q i plus delta L over delta q
dot; remember these are independent variables that was our big lesson motion in free space
and delta q dot times dt. Time is not a dynamical variable time is the
arena in which the dynamical variables vary in this way of looking at things. So, there
is no question of varying time, there is no delta t and the point is that you assume you
are looking at only those variations, where this delta q here and at this point as 0 in
between you allow all possibilities. So, this is the equation I will get, but then
remember this delta q dot equal to delta of d q over d t; and the operation of differentiation
with time has nothing to do with shifting this q. The fiducially shifts and the q those
are not dynamic variations those are just arbitrary shifts, arbitrary variations. These
two operations have nothing to do with each other, therefore this is the same as d over
d t of delta q. These two quantities commute then nothing to do with each other. And once
you put that in this equation of motion reduces to t 1 t 2 d t and inside the bracket you
have delta L over delta q plus delta L by delta q dot d over d t this is it.
The next step is obvious, I would like to pull out sorry there is a delta q here, I
would like to take this out as a common factor, but there is a d over d t sitting there. So,
the obvious thing is to do is integrate by parts. Therefore, say 0 equal to t 1 to t 2 dt delta
L by delta q minus d over d t delta L by delta q dot. Now, I can take out a delta q, because
I integrated this term by parts and put the derivative on this, that is the term I have
written here and you get a minus sign here of course; plus the boundary term and this
boundary term is of course, plus delta L over delta q dot delta q evaluated at t 1 and t
2. Because, you have the surface term because
of the integration being finite integration here you have the surface term, evaluated
at these two points. But by definition delta q is 0 at the two end points; two end points
there is no variation therefore, this is 0 by definition. And went up like this and this
is supposed to be true for any given t 1 and t 2 for arbitrary delta q chosen as you please
that is only possible if what is inside the curly bracket itself vanishes.
So, since this is true for all, since this for arbitrary delta q, we must have delta
L over delta q sorry I have to put outside, I do not need that here minus or equal to
d over d t d L by d q dot. Let us restore the i that is the equation of motion that you get.
Now of course, there is many, many questions we would arise the first one is why should
I take the Lagrangian what ever it be. I am going to give you subsequently, why should
I take it to be a function of q 's and the q dots why not the q double dots q triple
dots and so on. Now, my statement is that, we are going to
assume that the dynamics is described occurs in free space and the independent dynamical
variables for any system are the generalized coordinates and the generalized velocities;
and everything else is derived from it. Newton's equation already tells us this is true because,
Newton's equation says something about the acceleration and that is determined by the
force we apply on a system. On the other hand the position and the velocity can be specified
by you independent of what force you apply. Therefore, they are the independent variables,
independent variables are always which you are free to specify; everything else is determined
by the rule of the game. Therefore, that experience has been taken over, generalized to many many
other situations and it seems to be true in most cases that you can think of and in those
cases where it is not true, you can answer why it is not true, because you have not included
the right number of variables. There are some few exceptional cases, where double dot also
as an independent variable, but in most cases it is just q's q dots.
When we study Hamiltonian dynamics we will see this in a little more deep fashion, why
its just q's and q dots positions and momentum not anything more that is the way it turns
out to be. Of course, mathematically you could allow for q dots and q double dots and triple
dots and so on. If I had a q double dot for example, L was q q double dot then there would
be a term delta of q double dot. Again I would remove the time dependence by integrating
by parts I would have to do it twice; and if you did it twice you would end up with
a plus sign and plus sign would lead to a plus d 2 over d t 2 delta L over delta q double
dot and that would sit in the equations of motion.
You could go up to n derivatives, any number of derivatives and they would alternate in
sign. This does not happen whole lot of times, so we are not going to look at that there
are some classical problems, where it does happen where there is dependence on the acceleration
as well. But as I said before the majority of systems do not have this behavior. These
equations these are the equations of motion if you are going to deal with very very important
they are called the Euler Lagrange equations. Next thing is to tell you what is L that is
million dollar question, what is L? Now, as I said we have to write down L, this
is the whole point for a given system I have to write down L, I will do this by experience,
but we will use this we will use a trick. The cases which, we already know the equations
of motion Newton's equations I am going to tailor an L to produce those equations
of motion. Then, I will generalize after that and it
turns out that in a whole lot of problems, conservative problems without dissipation,
L turns out to be the difference between the kinetic energy and the potential energy. This
is not a formula for L; it is just one possible expression for L which yields the correct
equations of motion. And there are other cases where it is not true and we are going to do
one very such important example. So, let me write down what L is in such cases
will check out this is really true or not, so in the simplest cases, without friction
without dissipation. L turns out to be T minus V just check this out whether this is true
or not. So, we imagine a collection of particles moving in some potential there is mutual interaction
between them, there could be an applied external force and so on do not care.
But a certain L is T minus V in this case now what is T normally lets look at a case
of a set of particles whose coordinates are specified by q 1 q 2 q n. The kinetic energy
is easy to write down, so what would T be in such a case lets take a very simple case
where we assume that the q's are actually the Cartesian coordinates of a set of particles
moving around. Then, this would be equal to one half m i q i dot square summed over i
that would be the kinetic energy, set of particles minus potential in general would depend on
q 1. Good question, L is Euler function, L is a
scalar function, it is not a vector. Pardon
I am saying it, I am asserting L is a scalar function, the kinetic energy is a scalar function
right. It could have had a vector function absolutely,
but L is a scalar value. So, I should have defined it L is a scalar value. It has the
physical dimensions of energy back to I state. We will generalize it as we go along we will
see what is going to happen and we will also see how we are going to write down L in more
general cases; that is really the trick when more complicated situations what is the possible
what are the generalizations where does this get us this is a crucial question we will
answer that. So, in the simplest of instances if I simply
assume all the q's are the Cartesian coordinates of various particles moving around, then this
is what the kinetic energy looks like and this is what the potential energy looks like.
What is the Euler Lagrange equation give for you it immediately gives you exactly what
you want? Because, delta L over delta q dot since this potential does not have any q dot
dependence we are assuming potential without any velocity dependence here.
Delta L over delta q dot is minus delta v over delta q i, that is the left hand side
of Euler Lagrange equation and what is the right hand equal to I have to differentiate
this and these q i's are all independent variables, the q dot dependence is sitting
here. So, delta L over delta q i dot delta L over delta q i dot is equal to the half
cancels out it is just m i q i dot no summation for each i this is true and what is d over
d t of this. So, it says the mass times the acceleration
is equal to the force this is the force. So, it reproduces Newton's equations. It will
do a lot more than that we will see, but it reproduces in the simple instance Newton's
equations. Therefore, to
use this formalism successfully, you are going to have to write down the kinetic energy and
the potential energy even in those cases, where L is T minus V and that itself can get
quite intricate as we shall see. But, in the very simplest of instances it
does this. Now, let me show you immediately the next advantage of using the Lagrangian
formalism. You know there are very many problems in which you have constraints, the coordinates
are independent of each other there are constraints and then the standard way which you studied
in high school was to use constraint forces like normal reactions.
If I want to study the motion of this object in this table, I better include the normal
reaction of the table on this. Now, this is a bugbear for generation of students, because
they really do not know what this normal reaction does its only purpose is to keep this object
on the table and if you ignore it you are in deep trouble whereas, at the same time
you do not really know it is a physical force or not certainly is, but this is a pain.
Keep these constraint forces in or if you put a particle on a bead or a string and move
it around its always on the bead, on the hook, the bead is always on the hook and there is
a normal reaction, reaction force, this is difficult to keep track of. Let us look at
the simple problem where you have a constraint of this kind and I will show you the Lagrangian
formalism actually present involve the constraint forces explicitly.
Once you eliminate coordinates it takes constraints into account automatically and this is important.
Let me go back and the risk of causing that little sinking feeling in the stomach take
you back to JEE and will look at airports machine. This is something all of you have studied
and its just a simple pulley of this kind and there is a mass here and there is another
mass here M two and these guys are pulled down by gravity; and this is supposed to be
a pulley which is frictionless and so on and you are asked to find what is the acceleration
of the mass M 1. Hold these two guys and let the heavier one
will sink and you are asked to find its acceleration, what is the formula?
Pardon me you have to assume a tension in the string clearly its been a while since
you answer because three years junior to you would write the answer down instantaneously.
Pardon me G divided by M 1 plus M 2 well the acceleration downwards of the heavier mass
is the same as the acceleration of the upward mass, we need to know what it is?
Difference over sum multiplied by G that is a good way of remembering, let us write the
Lagrangian down for this. We need to write some coordinates we are all interested in
vertical coordinate. So, let us put the origin here and let us say the position of this is
at x 1 and the position of this is at x 2. I measure x downwards and I take the potential
energy to be 0 here. So, that this potential energy is minus m g x 1 and this is minus
m g x 2. So, the Lagrangian as you would normally write
down T equal to one half m 1 x 1 dot square and the potential energy v is minus m g x
1. So, L is T minus V, but now I have a constraint the constraint says that there is a tension
in the string right at every point you assume that there is a T upwards and a T downwards.
Now, what is the purpose of this constraint, what does it really tell you.
Pardon me the length of the string is constant just say the string is inextensible that is
it, in that sense incidentally I am going to give you as an homework problem what happens
if the string is not inextensible, but actually can extend then of course, this constraint
is not used in the way is simple fashion. But let us write down what it is when the
length is constant if the length is constant then I know that x 1 plus x 2 equal to some
l or x 2 equal to l minus x 1, which immediately says x 2 dot is minus x 1 dot that we know.
We know that the speed of 1 is minus the speed of origin. So, we remove the constraint by
putting into this, then L becomes 1 half m 1 x 1 dot square plus 1 half m 2 so, we add
up minus V. So, that becomes a plus m 1 g x 1 plus m 2
g x 2, but x 2 is L minus x 2. So, you permit me write this as m 1 minus m 2 g x 1 and then
there is a minus plus m 2 g l. This is irrelevant, because it is a constant you can always add
a constant to the Lagrangian and since the Euler Lagrange equations do not involve derivatives.
If you add a constant to the Lagrangian it does not get differentiated on either side.
So, the first lesson the Lagrangian is not unique. You can add constants to it its not
going to change any physics, but this you already know; the potential energy V of q
you can arbitrarily shift its reference level and it does not matter. You can do all these
tricks non relativistically, but you cannot do relativistically.
Because, when there is something called absolute zero of energy then you cant shift things
around, but in this kind of non relativistic physics you are not worried about rest masses
and so on. Then the zero of energy can be shifted as we like you can even shift it by
infinite amounts in the simple harmonic oscillator you take the minimum of the potential to be
0 at the origin and then the potential is half the x square going up.
But in the hydrogen atom problem or the Kepler problem, you take the potential to be minus
1 over r, and you take it to be 0, when the two particles is infinitely far away from
the center of attraction. So, you really move the 0 to point at infinity. May be problem
you have to specify before hand, what is your reference level is now we specified r as potential
energy being 0 at this point. But it is irrelevant and that is why it comes
out as a constant and now we short order the equations of motion delta L over delta q i
there is only one q lets write and it immediately says m 1 minus m 2 g is equal to d over d
t of M 1 plus M 2, so that comes out i differentiate L with respect to x dot and I get a x one
dot here two cancels and I do d over d t x 1 double dot, therefore the acceleration.
So, I did not introduce a constrained force. On the other hand I use the constraint to
get rid of one of this coordinates we will see with further examples, this is one of
the real advantages; the real power of the Lagrangian method you do not have to explicitly
introduce constrained forces. You can actually use the constraints to eliminate coordinates,
when ever you can and then automatically the equations of motion to the remaining independent
coordinates emerge; of course x 2 double dot is the negative of this.
Now, the deep question what happens if you put in friction then of course, you need a
model for the friction, imagine both these things are suspended in some liquid and then
there is some friction which is proportional to the velocity you cannot write down such
an easy Lagrangian. We will see in a minute how to do this in
some cases in all in a general case there is no prescription as such really is not really,
do not know the dissipation either classically or quantum mechanically, simply because the
nature of this is so intricate the nature of this dissipation. The mechanism has to
be identified and you have to write a model for it. So, that is a very satisfactory answer
at the moment we will look at examples. That is again a deep question where does friction
really come from all forces are conservative, where does friction let me give a quick answer
of top of my head, where does this friction really come from ultimately if you have a
surface here and another surface and you do this the friction comes due to molecular forces.
Surfaces are rough of course, at the molecular level they really are not uniform surfaces
at all and then bonds break, so these are really at the molecular level. So, first we
have to use quantum mechanics for that, but ignore that for the moment really is at the
molecular level. But, then this means the energy can be transferred
from one system to another at the molecular level, what is molecular random molecular
motion called it is called heat. Really it is entirely possible that even though in principle
all the forces that you deal with are conservative forces; it may turn out that some of the work
goes into heat into random molecular motion which you cannot recover.
The origin of irreversibility is very, very deep, when we do statistical mechanics I will
make many more comments about it, where this comes from what is the way we understand it,
where does macroscopic irreversibility come. Although the even though the microscopic atomic
and molecular level you have reversible forces, this is a deep question one of the real unsolved
unanswered questions; in it is full we have lot of answers for the last 150 years.
But, it is not clear that the question is completely settled.
Good the question is, is the Lagrangian formalism is just a tool for solving problems does it
explain anything more than Newton's laws? The answer is yes indeed it does much more
than Newton's laws, so let me list some of these things and then we will come back
try to get this straitened out. First as a problem solving tool, I have already
mentioned that it takes constrained forces into account, that is the first advantage;
the second advantage it has is that its possible to generalize it relativistic mechanics, when
Newton's equations are not valid. So, the Lagrangian formalism will give you the equations
of motion the right equations of motion in the relativistic regime as well.
Then, the Lagrangian formalism can be extended to the case of a continuous number of degrees
of freedom. In other words fields which is not possible with Newton's equations of
motion. Newton's equation of motion, do not look anything like Maxwell's equations
of motion for the electromagnetic field. But, Maxwell's equation of motion for electro
magnetic field, or the Euler Lagrange for the electromagnetic field written down from
Lagrange. So, again that unification exists, so its
possible to relativistic physics its possible to do fields and its possible to include constraints.
These are among the advantages that go well beyond Newtonian framework. Disadvantages
not drawbacks in the sense that cannot be applied or anything like that, but disadvantages
not easy to quantize when you want to quantize systems.
Then, the Hamiltonian framework which is like another brand of tooth paste, if you like
this works we will see the equivalence between these two . One of them can be pointed out
right away, what is the order of this differential equation; second order it is d over d t and
there are q dots here, second order whereas, we kept saying the dynamics is going to occur
in free space and everything is going to be first order differential equation.
So, you should have taught me immediately that these are second order differential equations.
We got over it by saying each time, I defined x dot this v n then I put v dot equal to the
force divided by the mass, but really these are the Euler Lagrange equations and the second
order dynamics Little more complicated then the first order
dynamics, we used to that is why we will shift to the Hamiltonian formalism which is first
order dynamics and these are equivalent. So, it has that extra advantage.
To understand where Newton's equations came it was just written of the top of Newton's
head right, so you wanted to understand where did it come from it turned out there is a
unifying principle, all the principle of least action a little bit of history many people
worked on this. Many, many people worked on this, all the
big names are Hamilton, Jacobean, many many people worked on this before they got the
correct equations. I agree, but then I have to tell you i am
going to give you an answer which may not be very satisfactory, but that is the only
answer there is, this is not an axiomatic subject we haven't reached a stage where
something some absolute truth is written at some level and everything is derived axiomatically
from that physics does not work like that. It starts by simple examples and generalizes
and once you put whole lot of things together and the same formula applies for all of them
instead of individual formulas; you take that as granted and then you ask what are the limits
of its applicability and generalize it and further and further and further and so on.
This is the way it goes and that is how we found that all these equations of motion can
all be subsumed by one principle, principle of extremal action. So, we believe it is true
till somebody comes with a better alternative. So, believe me this has been tested for several
100 years and this is at this level it is completely established that where are you
going to write the Lagrangian from. Especially, when you go to quantum mechanics
when you go to quantum fields where are you going to write the Lagrangian down from you
do not have the experience of Newton's equations you do not have this. You are not in the non
relativistic regime you are not in the classical regime then invariance principles play a big
role. It turns out nature is guided to a large extent by symmetry and invariance principles
which already manifest here which already are hidden here.
But, I bring them as we go along you would write the Lagrangian down based on considerations
of simplicity; simplest possible choices subject to the invariances one of them is he already
pointed out the Lagrangian must be a scalar; but, not a scalar in the sense that you and
I understand the scalar in this course, whereby scalar means something which is not a vector.
Something does not change under rotations of a coordinate system, but by scalar something
that does not change not only under rotations of the coordinate system, but also Lorenz
transformations shifts from one initial frame to another. I will use what is called a four-dimensional
scale or a Lorenz scale, scalar under special relativism that would guide me in writing
down the equations of motion in more complicated sequences.
So, this is what we are going to do start with the simplest and go on to the more complicated
problems. The very next problem I am going to look at several of them I want to do, but
let me do one of them at least is the problem of the simple pendulum and this is again something
we will return to just to give you a inkling of how you should handle coordinates other
than Cartesian coordinates. So, let us look at the example of a simple
pendulum. Pardon me
The least action principle Yes, yes.
Well that is that is also an interesting question we derive the equations of motion by looking
at those variations, where delta q vanished at the two end points; you could ask suppose
I do not do that I look at other variations would I get a different set of equations.
The answer is no you get exactly the same set of equations, so, I chose the simplest
set of principle to derive the equations you can generalize it.
You can look at arbitrary variations and Euler Lagrange equations do not change. So, that
is it? Yeah yeah
I think what he is trying to say is a very very interesting question once again. It looks
like time t 1 here time t 1 tomorrow and now integral from t 1 to t 2 is what is being
minimized. On the other hand the system does not know what is going to happen tomorrow
right now. So, how does it do it and that is the reason
we end up with a differential equation. So, it is really a local object its going from
one instance to another. Remember the action principle was true between any t 1 and any
t 2. So, you could slice up any time between t 1 and t 2 into intermediate intervals and
for any two of them the motion would again be along the action path. So, that is the
way the differential equation. So, let us get a few more examples under our belt before
we discuss further formalism. So, let us look at the pendulum what does
this do I have in mind a light mass less rod of length l, a bob of mass m and this bob
executes oscillations in this fashion. We will assume that this is the given plane and
it is frictionless and the oscillations occur with some angular displacement theta and I
will take this to be 0 theta is in the vertical position.
So, what is T in this problem this is a light mass less rod. So, the only mass is here the
kinetic energy is just the kinetic energy of this bob. And what is that equal to
l theta dot is the linear velocity, so half
m l theta dot square is the kinetic energy. And what is the potential energy? Again let
us assume the potential energy under gravity to be 0, when it is at the lowest position
here; and then at a given rate this is the amount by which it is raised. And therefore,
it is m g times the height there and v therefore, is m g l into 1 minus cos theta right. So,
the Lagrangian is equal to one half m l square theta dot square minus m g l plus cos theta. And all the Lagrange equation of motion says
delta L over delta theta which appears only here is equal to d over d t of delta l over
delta theta dot, which will give you m l square m l square d square theta is equal to the
derivative of, which is equal to minus m g l sine theta or theta double dot is equal
to minus, that is the equation of motion of a simple pendulum. That is the exact equation
of motion are these harmonic oscillations are these harmonic oscillations is it simple
harmonic motion? No, only for sufficiently small theta when
you can approximate sine theta by theta then of course, it is the harmonic oscillator problem
with a natural frequency with square root of g which is square root of g over l and
the time period is 2 pi root l over g. Otherwise it is not true is this a linear problem is
this equation of motion linear highly non-linear, the sine theta here is all powers of theta
from 1 to infinity. How small should theta be in order that you
make this liner approximation, how small should it be 4 degrees, 6 degrees, 0 of course, then
of course, it is exactly true. He says 0 I think that is pretty funny right, so we are
going to laugh at him for a while then we find that the laugh is on his side of friends,
how small should theta be absolutely I am very happy to hear that this class does not
give me the answer that should be 4 degrees some other state its 6 degrees because some
guy wrote a text book saying that it should be less than 5 and a half degrees and so on.
This is meaningless it depends on your accuracy you are after all neglecting theta cubed over
six compared to theta in a sine expansion and therefore, you can easily find out how
small should data be in order to precise you specify an accuracy and I tell you how small
theta should be. If you want 1 percent accuracy it has to be much smaller than if you want
10 percent accuracy and indeed if you want complete accuracy you have to do what he does,
when we put theta equal to 0 only then these two are exactly equal. But, otherwise it is
not a linear problem, but for sufficiently small theta some prescribed degree of accuracy
this becomes a linear equation, then you can solve this. This equation as it is also is solvable in
explicit closed form, but the solutions are called elliptic integrals; and the time period
increases as the amplitude increases and like the cases of harmonic oscillations. This equation
is very profound it occurs in many, many parts of physics not with g over l some constant
here it is called the sine god in equation simplest equation.
But, do not look at it right now what is the phase portrait of a simple pendulum going
to look like, what is that going to look like. And let us quickly do that because remember
this is a rod its not a string it is a rod and I am going to assume that it can actually
go around all the way. So, you have oscillations and then you could also have complete rotation, What is the phase portrait look like well
at least theta dot plane; here is theta this is theta dot what would it look like. It depends
on where the critical points of the system are, where are the critical points of the
system? Let us write it as we should really write it in terms of two variables well I
would not do that for a moment I will come back and do that in the Hamiltonian framework,
but its clear that the speed theta dot angle of velocity is 0 at equilibrium and you are
at the minimum or maximum of the potential. What does the potential look like? By definition
it has the minimum at 0 and then it goes up and comes down. So, it is sine potential it
looks like this, this is 0, this is pi, this is 2 pi, this is minus pi, this is minus 2
pi and so on. So, it all at all the odd multiples of pi it has maxima unstable equilibrium points
and at the multiples even it has minimum. So, what do the critical points look like
there at every even multiple what do they have a centre, a stable center.
At every odd multiple what would you have a saddle point. So, this thing would be very,
very simple center, saddle, center, saddle and so on. And you would have small oscillations
about this point or this point or this point. So, this pendulum could have small oscillations
about 0; or it could have been rotated once and then about 2 pi it has small oscillations
and so on. They are all equivalent points you could have an oscillation which starts
at minus pi and goes all the way to plus pi, but does not quite till it over and then go
back. So, that oscillation would correspond to starting
a little bit going all the way up there and then going back and of course, as the restoring
force goes to 0, it is going to take longer and longer to crawl up to equilibrium. So,
you really eventually would have larger oscillations amplitude oscillations; and then a separatrix
and there would be a separatrix from this point to that and one from that to this and
so on. So, two guys going in two guys coming out,
so as the energy of the pendulum is increased if this is the energy you could have periodic
motion in this well or this well or this well and so on. Those are those closed orbits as
you go up to the critical value what is the value here what is the critical value of the
energy separatrix. It is the maximum of this potential 2 m g
l and that value you have these separatrix solutions this is again a cycle, but it is
not a homo clinic cycle which has two saddle points, connects two saddle points what would
you call this it is not homo clinic cycle it is hetero clinic cycle.
But, it is very similar to the kind of picture, we saw for a single maximum and minimum what
happens for energy greater than 2 m g l what happens if I give this much energy. Well clear
this fellow can escape over the barrier and make rotations; but of course, very time the
potential energy is very high the kinetic is low every time the kinetic the potential
is low the kinetic is high. So, you have this kind on this, this side
you have this kind what do these open orbits correspond to, what kind of motion? Rotational
motion rotation and what do the closed orbits correspond to? Oscillatory motion both are
periodic, but one of them is oscillatory and the other is rotational. Open trajectory E
is greater than E sub s rotational motion and this corresponds to E less than oscillatory
motion. What happens to the time period as you go
towards the separatrix, what would be the time period on the separatrix? Infinity absolutely
right infinity, can we show this in a simple way, it is actually infinite we do this in
a simple way. You have to write the formula down for the time period and you want to ask
how long does it take say from here to here. And the answer is going to be infinity because
it is going to crawl from this point up there and I am claiming that this has taken infinite
amount of time. Because, the time period T is going to be
proportional up to pi, theta over theta dot, we have to hit the value pi and you have a
theta here and a theta dot here and what does this theta dot look like. Well put e equal
to this whole thing is equal to is the energy, so one half m l theta dot square plus the
potential energy total energy and the total energy is two m g l or a separatrix and put
that in and you end up with a d theta over which something blows up when not integrable
over sine theta or something like that which blows up at this point, therefore, the answer
is infinity as you expect it to be, so we will check that out.
What is very interesting in this problem the pendulum problem for deep reasons is that
although you cannot write the actual function theta is the function of t in elementary terms,
when the amplitudes when the oscillation is no longer simple harmonic, when you have to
the full equation here I said these are elliptic integrals. On the separatrix you can once
again, so if you put E equal to 2 m g l you can actually solve for theta as a function
of t and it is related to what is called a Colton solution for the sine Gordon equation.
So, I am going to give this as a homework problem even the simple pendulum as you see
has very deep physics buried in it this equation here in particular appears over and over again
in applications huge number of applications in generalized form. We will try, we will
see, we can encounter some of these.
