Hello now we discuss one other small concept
but a very profound concept in calculus of
variations called general variation that leads
to a number of interesting things such as
what we call transversality conditions, broken
extremals by broken what we mean is actually
not made in two pieces but something that
has a kink that instead of having a smooth
curve there will be a kink such as this that
is called broken extremal.
As if you have a continuous wire suddenly
if you make a kink in it that you say wire
is broken and that kind of broken extremals
and we also look at corner conditions basically
those broken extremal points or corners physically
a corner in the curves we will discuss that
and that also can be extended to surfaces
that is one dimension two dimensions and three
dimensions it has important implications in
applying variational methods to mechanics
will consider some of those examples as well.
So let us begin with this concept of a general
variation and look at outline of this part
of the lecture so we will consider variable
end conditions that is what leads to the concept
of general variation and that leads to what
we call transversality conditions and also
this corner conditions named after we are
stress and Edmund actually there will be another
end here okay, what we learn here is how to
deal with variable and conditions and how
we can take this general variation which is
more than eager to variation that we have
studied.
So far and we see how broken extremal looks
like through examples so we will do we look
at some examples to look at this broken extremer
in fact one example that we would take would
be the very first problem we had discussed
which is the Fermat effect refraction okay
we can call it Fermat observation we can even
call Fermat effect because he is the first
one to explain refraction of light saying
that light takes a minimum time path and not
minimum distance path okay so that Is what
is our agenda for this part of the lecture
okay.
So as a motivation for this let us consider
the famous problem in calculus of variations
which is under the effect of gravity if I
have point A and point B we say if I were
to connect it with a curve and make a slide
out of it if I leave a ball here and that
ball will fall under the effect of gravity
you should take the minimum time that is what
we call Brachistochrone minimum time problem
that is a famous problem this famous problem
is solved more generally and that is actually
general variation that we need to discuss.
Now what we say is that let us say that this
point B is not there okay instead what we
say is that this point can lie anywhere on
this curve okay we can take some coordinate
system let us say x and y okay let us take
a curve called Ø x okay that is this now
we say we have a ball as it is shown over
there and say that that ball has to come to
some point on this we do not care which one
okay.
You just come and touch this thing under the
effect of gravity starting from this point
right there are many curves right so we can
go there we can go here we can go there so
there are lots and lots of possibilities at
another effect of gravity you make slides
in each of the cases whether string this ball
onto the wires you can just constrain to move
there or we can make a slide so that you know
goes and goes there which one of these will
the ball take rather which curve we can say
which curve if I say this is y of X right.
Which curve will minimize the time to go from
point A to some point in on f of X on that
given curve this is our given curve that is
the modification we do what is the consequence
of this consequence is that when you start
let us say these our x axis if I do this I
said this is x1 right and what is x2 we do
not know it could be this it could be that
it could be this it is variable right let
us say y star is a solution then this comes
x2.
But this x2 is variable meaning that we know
x1 but we do not know that and yet we have
to have the integrand with all its glory first
derivative second derivative function many
variables and whatever right so are many functions
when it have the functional we do not know
the bound right it need not be only the ending
thing it can also be the beginning thing one
of them we do not know is that is our real
thing here we may not know this we may not
know that we should be able to or maybe both
of them we do not know that is also possible
we can makeup problems where both of them
are not there okay, that is a modification
of the brackets token problem.
That is what is shown on this slide we specify
point A all right but B can be anywhere on
this curve and hence we have a variable condition
of this x2 not specified so this is X 2that
we do not know we have to find it as part
of the problem you can actually see the generality
now right because we are not just minimizing
the functional that we have but if the functional
is an integral the bounds are not known okay.
That is the modification of the blackest of
one problem okay were Ø2 as a function is
given here where point B has to lie right
and this is the time taken this is the time
taken by the B2 go from point A2 some point
on the curve Ø 2x time taken to go under
the effect of gravity from A to some point
on Ø of x we do not know what B is that is
an unknown well that is what we will solve
today okay.
And if you wanted to make it even more general
as I said I can have one function there let
us call that Ø 1( X) again choose coordinate
system this is x-axis and y-axis is Ø 1 we
can also have another thing Ø2 okay let us
say this is Ø2 that means that my integral
of the time taken I would say x1 x2 but I
do not know what x 1.is I do not know what
x2 is both are variable because I can start
from any point on this several points right.
Many, many points on this and end up at any
other point so now our such space has really
gone so I can start from there and go here
to go there or start from there and come here
and right so possible curves that we have
are too many right out of all of those things
we should pick the one that means this T this
is the variable and conditions general both
end conditions are very okay and that is what
we have shown here.
S this v Ø1x, Ø2x are given and we need
to find a point A that lies on Ø 1x point
B that lies on Ø 2x and minimize our time
were both the ends are not known in fact whether
one is known or both are not known does not
matter one is not known both are not known
does not matter once you understand how this
type of problems can be solved okay.
And so our general variation that is needed
for the problems which have variable and conditions
looks exactly like this the change is only
in the ends those are variable ends we do
not know what those are where to write them
okay for that we need to do this general variation
concept along with why we should arrive at
x1 * x 2 *
So now we need to find so far just a function
we have to find a function which we denote
as y * X but here we also need to find x 1*
and X 2* if they are not given sometimes one
of them may be given other is not given but
sometimes both may not be known you both are
given that becomes a normal problem. If one
of them is both of them are not given then
we have this concept of general variation.
What is general variation we talk about let
us discuss that first so let us say we take
our x axis and this is our y ax is the function
that we are trying to determine right so let
us actually call this X 1and X 2 so let me
actually draw a curve which might be say from
here till herein some form okay that is what
we call y of X which may be the solution y*
also okay now in order to check if that is
let us call this Y* which is the solution
that minimizes a functional right.
But now in order to check whether it is actually
the minimum we always been doing perturbation
right that perturbation is this Y * we need
to part of it so that it is different now
since X 1 and X 2 ends are also not known
we have to put them part of them also let
us say this becomes x 1 + d X 1 and X 2 becomes
x 2 + d X 2 so d X 1 and d X 2 these are the
perturbations 
in what perturbations two let us call it they
are not perturbations in their perturbations
given to the ends.
Which we do not know that is why we need to
do see whenever we say local right from the
beginning of this course you have been saying
if somebody gives you an answer and say this
is the minimizing quantity a number or a function
we need to perturb it and see locally it actually
gives the least value objective function with
or without constraints so here also when we
say that this is a minimum it is not enough
to perturb y so Y * that we have right we
can give d Y for it perturbation right.
But we are not satisfied with that here because
end conditions are also not nodes were part
of the end so d x1 x2 are those perturbation
that means that I need to define a new function
let us say from here that one and from here
somewhere we have to do this the new function
has to go from here to here something like
that okay so from the red curve to blue curve
this is the perturbed 1 this is y * that we
have plus d Y right notice that the domain
of d Y has changed that this is the new domain
the old domain if you see the old domain is
from here to only here x1 to x2.
But now it is from x1 plus d X 1 to X 2 +
d X 2 that is what we should do and see what
happens okay it may look very complex at first
sight but once you understand what this general
variation that we talked about right once
you understand the answer is actually quite
simple it is a profound concept but is a very
simple concept right, we are perturbing the
domain as well as the function.
And that is what is shown here and we have
a very fancy name to it called non contemporaneous
variation okay what that means is that this
concert was dealt mostly for dynamics problems
we have the independent variable X was actually
time T right when you are saying that I want
to do from t1 to t2 your function defined
now we have t1 + d T 1 T2 + d T to write because
of that this non contempt that you are not
do in the same time right it is not contemporary
it is non contemporary that is what this is
general non contaminants variation so we have
our Y * that is given and then we have the
perturbed one with this Y * +HX in our terminology
our d Y both.
So H is d Y we have been interchange in that
basically that is the variation here we call
it general variation this is variation we
are called general variation because of change
is the domain also so we take this d J to
see after perturbation the functional value
increases or decreases or two first order
it should be equal to 0 that is our criterion
to first order the function should not change
and then we will say whether it is low minimum
or maximum so we do from d x 1 + x 1+ d X
1 to X 2 + d X to subtract from it our usual
functional both original domain okay.
And that is what we have taken but we rearrange
a little bit so that our limits are x1 tox2
for the functional with H and without H if
you look at this integrand okay let me change
the color of the ink okay if we look at this
over there and this over here both have the
same domain but we know that originally this
one that is integrand the depends on the function
and the perturbation okay that one we have
it was from d X 1/ d x 1to x 2 + d X 2 that
is from here to here so we need to subtract
and add so that it goes from x 1 to x 2 okay.
So this part is subtracted because this is
included when you two when you go from X1
whereas it is starting from X 1 is d x 1 so
X 1 to X 1 d X 1 is subtracted and x 2 d x
2 was there here okay originally we have this
for further that has to be explicitly added
so that everything is the same now is the
critical thing very simple but very important
one if you look at these limits it is going
from x 1 to x 1 + d X so that means that it
is going over a very small distance d X 1because
small distance d X 1 we can simply write that
portion as f value at X 1 times d X okay so
this particular integral that I am underlining
is nothing but this little term here.
Because the integrand is not going to change
much because you are going to make d x1 a
perturbation d x 1 d x 2 as small as possible
that becomes F evaluate that point x1 times
that and likewise the second one we take a
different color and say this one right that
basically amounts to this term because in
that little range from x 2 to x 2 / d x2does
not change it is enough to take FA at that
point time d x two. Other two things the first
and second term that is what they areas it
is that are in this line is usually what we
had earlier okay that is the difference.
One other difference if you see what we have
here is the age that we are writing okay I
just said an earlier slide that age is d Y
but here there is a small difference okay
the difference is what we will find out now
right if you see what is perturbation H versus
what is perturbation d Y okay let us understand
d Y if I say at the left end d y1 it is from
a point that is here to the new point that
is there okay that is d y1 but what is actually
h that is H will be the between the two functions
if I take any point here that will be H okay.
That will be H here whereas d is the corresponding
points the contemporaneous we can call it
that way because this points actually correspond
to this point by the end right again this
point okay is correspond to this phone if
you put arrows so this point and this point
we are comparing that is d y1 whereas at the
same point if I take and same one of the curve
that is H perturbation the function where
a delta y is actual variation including the
domain condition there okay.
So we have now delta and d y 2 and those relationships
are given here by extending the curve if you
see we do not have anything here if I want
to know what is H here at this point x1 what
is H we do not have a point there does not
come so we have extend the red curve like
this right I am or blue I am writing this
extended curve the same slope and that is
what is given here y 1’ times ? X 1 similarly
Y to parent ? X 2 because we are extending
that a little bit over there like that because
we do not have want to say what is H there
and what is ? y1that we have from here to
here it is h1that is from here to here that
is h1plus that part which is y 1 ‘ ? X 1.
Similarly ? y 2 is h2 plus y 2 ‘ ? X 2 and
that is what we will use when we try to solve
this for now we have h1 h2 we replace that
in terms of ? y1 and ? y 2 okay if we do that
what we end up getting is our then we have
to do the usual thing that is go to variation
equivalent here when we have this and this
okay f y star Y ‘ star we have Y sub y h
y, y ‘ H ‘ okay and in order to get rid
of this H ‘ we do integration by parts that
gives us the differential equation part of
it integration by parts will give you d by
DX the usual thing or you also get the boundary
condition okay.
And that is what is shown here with the boundary
condition put at x2 and x1 now what we do
is for this H we replace in terms of our ? Y
and Y ‘ and so forth right if we substitute
there and also remember that ? Z ? J that
should be equal to 0 had these terms also
okay now overall what we get in ? J substitute
everything boundary conditions come these
are the usual boundary conditions is a special
boundary condition that we got because of
the perturbation of the domain itself that
when we substitute what we had for ? y 1 and
? y 2 in terms of h1h2 what we end up will
be this integral to which we can apply fundamental
m of calculus of variations because H is arbitrary.
And hence we get Lagrange equation as it is
but now we get perturbation in ? Y and perturbation
? X okay because additional one comes y ‘ because
of these terms right this notice that now
we have perturbation ? Y function and domain
independently we get boundary conditions were
both okay.
And that is what we have we get in the function
perturbation this is domain perturbation if
domain is not perturb it is fixed let us say
that endpoints are given x1 x2 are given then
this will be 0 they can be satisfied so that
is why it is called in general it is valid
for this and the specific case we discussed
so far in collective variations where ends
are given when ends are given ? X is zero
and so it is satisfied if it is not given
what will be 0 is that f minus f ‘ x y ‘ is
equal to 0 okay we can actually get it out
from the general thing we can get the specific
one that we have discussed so far.
And for that boundary condition is exactly
what is shown here though f by ? y ‘ which
is that x ? y equal to 0 Y is specified that
is 0 Y is not specified that is 0 okay that
remains the same okay now so far general variation
we said X 1 X 2 just not known whereas now
we are saying they have to lie on some curves
fee one on this side fee one on that side
v two on this side what does that mean that
means that when you take this into account
and observe these relations right ? y1now
cannot be anything because he has to satisfy
between ? X 1 ? y1 we have the derivative
of fee 1 and then ? y 2 and ? x 2.
We have derivative of V 2 unit substitutes
earlier this boundary condition this boundary
condition will separate now we can replace
? Y in terms of that and for both ends then
we get separately forx1 and x2 see that one
and two together let me change the color of
the ink yeah one and two together become of
course these were from x1 to x2 right now
we have written one condition another condition
X 1 X 2 because we have two different curves
so here the curve involved was fee 1 X right
here the curve involved is fee 2 x okay one
and two actually were at x1 and x2 there were
actually four conditions two conditions here
two conditions here that becomes actually
only two conditions overall okay.
Here 2 plus 2 total four conditions are there
x1 x2 this condition X 1 X 2the other condition
for but now we have need to because we have
found a relation between ? Y ? X because once
I am here if I have to be here when I move
when I move ? x1 I also have to move ? y 2
so those are constrained okay let me draw
a little bigger may be somewhere here if I
move from there to here if I move so much
at necessarily move ? Y okay that is what
are the relations okay and these are called
transversality conditions right.
Why the name we will examine all right so
what the conditions look like that when the
end is fixed then ? x is 0 then this must
be 0 then this can be need not be 0when it
is not fixed that that must be 0those are
the transversality conditions we have we understand
what those okay so transversality actually
comes in the following way if we take a problem
our integrand is of a particular form because
historically and that is how it got developed
the integrand here is of this form f of y
and then y ‘ has a specific form of square
root of 1 plus y ‘ square as you see this
you recognize that basically d/s or the arc
length on a curve.
That is square root of1 plus y ‘ square
DX squared DX square DX square plus dy square
square root when you take DX out that gives
you1 plus y ‘ square and the F ‘ FF of
Y can be anything if either that form now
we substitute into the transverse condition
that we derived in the last slide if you substitute
for F Y ‘ because that is what we have then
we get F which is simply F times square root
of 1 plus y ‘ square and here we have ? f
by ? y ‘ that gives you this if you simplify
as it is done here you would end up seeing
this Y ‘ fee ‘ is equal to minus 1 what
does that mean we are looking at fee of X
curve and then y FX curve their derivatives
at that endpoint are their product of the
derivatives is equal to minus 1.
That is true other curves which are perpendicular
to each other so if I have one curve let us
say there is another curve let me take a different
color for it that is I have another curve
like this at this point okay it is90 degree
slope ok there are orthogonal two curves at
that point right how do you say we have to
draw a tangent to this and let me draw the
tangent to this in its color tangent to here
between those two we have 90 degrees so are
orthogonal that is the thing that is their
transversal right that transfers to each other
that is why the name transversality comes
in okay.
That is transversality alright that is only
for specialty but the name has stuck as it
happens in, in any field somebody calls it
something that names get stuck and here even
when the integrand is not of this special
form we still call them transversality conditions
okay.
And four brackets talk alone if I take fee
1 x and fee 2 x then what you would know is
that if I choose the point a then it has to
be perpendicular to the curve there, there
should be 90 degrees
and likewise at this point also should be
90 degrees whatever point you take go there
ok transversality here exactly matches because
you will have y ‘ in the same way here okay.
And let us look at an example it is a an example
that requires little bit more work for us
to apply what we have discussed there is a
general variation consider a beam from mechanics
it is our, our normal beam we have fixed one
end right that is no problem the cantilever
connection right side we are saying that there
is a function fee to X here I am showing like
a line but can be a general curve also like
that fee2x and this right end of the beam
can be anywhere this is a physical problem
right you have put something and you are propped
on the surface as you apply the load it is
going to slide on it right.
So what is x2 for this beam we do not know
because if the beam bends like this that is
from here to here only right in another hand
if the beam if I use a different color another
hand if the beam bends let us say like that
then the domain is from there to here so variable
and condition so we do not know that end point
we do not know that but the difference here
is that we have now y double ‘ we discuss
the previous slides when the integrand has
y ‘ we discuss now to discuss what happen
is y double ‘ what happens when you try
triple ‘ and so forth okay which is what
we will do in the next part of the lecture
thank you.
