Hello friends welcome to on the todays lecture
In todays lecture we will discuss Uhh the
concept of calculus variation in fact this
is a first lecture of calculus of variation
So here in this lecture we will try to discuss
what is calculus of variation and what we
tried to do in in this calculus of variation
and how it is related to your usual classical
analysis
So here to start with let us consider the
Uhh this definite integral Uhh the Iy and
this is a definite integral from x1 to x2
f of xy dy by dx and this integral is with
respect to d of x and we know that this is
a welldefined quantity provided this integrand
f is a given function of xy and dy by dx and
here y is given function of x and this will
give you a number provided this x1 and x2
are having some kind of definitely numerical
value
So we say that that in calculus of variation
the first problem Uhh system find out say
Uhh the value is corresponding to different
different functions y and we compare Uhh the
values of these definite integral corresponding
to the given Uhh function y So we can say
that the first problem of the calculus of
variation involves comparison of the various
values assumed by this integral denoted by
1 when different choices of y as a function
of x substituted into the integrand of 1
It means that we have a value if y is replaced
by some y1 and then if we replace this y by
some other function say y2 then we may have
2 different values of this integrand and in
calculus of variation we try to compare these
2 values and try to find out a function for
which the value of this integral is either
maximum or say minimum or you can say that
here we want to find out a particular function
y which is a function of x that gives value
of this Uhh different integral as its minimum
or maximum value
So it means that once definite integral s
defined in terms of Uhh say functions of a
Uhh this another function y of x We try to
find out say a function of y in a way such
that this will achieve its maximum minimum
value right So in calculus of variation we
will discuss this kind of problems in detail
So Uhh so integral is of variation with try
to consider problems which can be transferred
into this kind of definite integrals
So here there are certain problems and some
problems includes the problems of shortest
distance between 2 points on a given surface
Uhh what it means that suppose we have a surface
Uhh it is like a you have a surface here and
then you try to find out Uhh that there are
2 points A and B and then we try to see that
what is the shortest distance between these
2 points Provided that these 2 points lie
on given surface
So this is one of the problems discussed in
calculus of variation and we will discuss
this problem in detail as the problems known
by Geodesics So such curves which gives the
shortest distance between A and B are known
as geodesics that will discuss in Uhh due
course of time and the second problem very
important problem is the Uhh the quickest
descent between 2 points
So it means how we can say Uhh consider this
problem That suppose in a vertical plane we
have 2 points A and B of course they are not
on a same height and then we try to find out
the curve connecting these 2 points A and
B in a way such that the if we take any particular
Uhh say with start from the point A and it
falls under the influence of say gravity from
A to B then it will take least time from reaching
A to B
So it means that the problem Uhh of quickest
descent is the problem of finding the curves
Such that under which if we Uhh particle it
slides from A to B under the influence of
gravity will take the least time okay So that
also we will discuss in Uhh detail and that
is a problem known as Brachistocrone problem
and this is the the first problem is which
is supposed to be the beginning of calculus
of variation we will discuss this problem
also
And the one more important problem related
to calculus of variation Uhh discussed in
calculus of variation is the surface of the
revolution of minimum area It means that suppose
we have a Uhh say say one function say y which
is a given function of y of x then we try
to between say 2 points say x1 and x2 and
then what we try to do here we simply revolve
this curve around this x axis Uhh with the
angle 2 pi then it will generate a kind of
right Yes it is like this okay So here it
is x2 and then we try to check the surface
area of the Uhh revolving surface and we try
to find out the curve y of x such that the
surface area of the figure so obtained has
a least surface area that we try to discuss
in this
So here we try to see that we will discuss
Uhh these problems in detail that all these
problems Uhh and the reduced in finding the
maxima minima of the Uhh (dif) definite integral
posts like this one so all these problems
can be termed as or can be written in terms
of Uhh some kind of definite integral which
depends on the Uhh say some function y and
we try to find out say minimum and maximum
value of these definite integrals and these
details or these methodology is closely related
to the problem of maximum and minima in the
elementary differential calculus or I can
say that techniques of solving the problems
of minimising or maximising of different Uhh
definite integrals even in 1 and related definite
integrals are closely connected with the problems
of maxima and minima in the elementary differential
calculus we try to see how it is
So now Uhh these Uhh definite integral are
termed as a name which is known as functional
So what is functional A functional is a mapping
which assign a definite number to each functions
belonging to the same class is called functional
so if you remember we have written down our
Uhh definite integral like this it is from
x1 to x2 f xy and y dash d of x So it means
that if you take a value y here then the corresponding
Uhh value is given by I of y axis
So it means that for any given function y
there is a value given by I of y So basically
it is a kind of a correspondence between a
function to a value so it means that it is
some kind of generalisation of a usual function
so we can say that that a functional is a
kind of function which say correspond between
y to f of y to I of y So here we can say that
Uhh that a functional is a kind of function
where the independent variable is itself a
function or a given curve
So here functional can be generalized as a
generalized version of function So it is also
a functional the only thing is that independent
variable is already a function of some Uhh
function of x So functionals are often expressed
as definite definite integrals involving functions
and their derivatives Uhh so in general you
can define any correspondence between Uhh
functions y to the value I of y
And so there are certain ways you can define
I of y as say Uhh this will go to say you
can simply say you may define it like this
also y of x not where x not Uhh is some point
in domain of say y so here also this also
represent your functional but here in calculus
of variation with try to consider the functional
which are termed as or which can be written
as these kinds of forms okay So here we will
discuss that functional are often expressed
as definite integrals involving functions
and their derivatives So in general we try
not to discuss this kind of functionals okay
So Uhh and the study of functional plays a
very important role in many problems and analysis
mechanics geometary geometrical optics theory
of elasticity quantum mechanics etc so that
we see as an application of calculus of variation
We will consider a certain problem Uhh of
these fields as an application of calculus
of variation that will shortly we will discuss
For the beginning we can say that the arc
length l of a plane curve connecting2 given
points Uhh is also functional what is this
So initially if you remember if we have surface
if we have a surface and we have 2 points
A and B and try to find out say minimum length
between Uhh minimum length of this curve connecting
these 2 points A and B So Uhh let us say that
these the surface is a planar surface or you
can say that these 2 Uhh points lie on a say
xy plane or yz plane or something like that
and we try to find out say minimum distance
of the curve
So here what we have Let us say that we have
a xy plain So here we have xy plane and then
we have a point A and we have a point B A
is given as x1 y1 So here this is your x1
and the corresponding is your y1 and here
we have say another point say x2 y2 So here
we have x2 y2 and then we try to find the
Uhh we can draw Uhh many curves connecting
this A and B and we try to find out that curve
which minimises the length of the curve connecting
this A and B
So here I can say that Uhh if you look at
the Uhh length of y it is given by say x1
and x2 and if we take this as d of s then
we can simply write d of s So it means that
length of y you can define as the integral
between x1 to x2 of this Uhh line Uhh patch
line Uhh element So here if we represent this
y Uhh curve as y is function of y of x then
you can say that d of s is given by x1 to
x2 and this is 1 plus dy by d of x whole square
d of x where x is lying between x1 to x2
So we can say that the length of the curve
Uhh connecting A and B a is given by this
functional and this can be simplified into
this x1 to x2 under root 1 plus dy by dx whole
square d of x and then this is one of the
functional we are discussing Now if we Uhh
we (consi) in in case of planer reason if
we consider these 2 point lying on a surface
then it means that in place of A and B lying
any plane
If we say that it is lying on some kind of
Uhh say Uhh some kind of a surface then not
only we have to minimize the we have to consider
the Uhh length of y but also we have to put
a restriction that these 2 points A and B
lying on a given surface
So the length of the curve connecting 2 points
is termed as a function of here So here we
can say that this L yx is given and x1 to
x2 under root 1 plus y dash square d of x
So here if you use one function U will have
a 1 value of this indefinite integral if we
use another Uhh function Uhh yx then we have
another value corresponding to that function
or you can say that given Uhh 1 curve there
is a different length
If we use another colour then we have a different
length so if we can use this then this is
y this is y1 So we can say here we have a
value or you can say length of Uhh the curve
connecting through this curve and we have
another length if we use this y1 as a curve
So we can say that this ly is a function of
your Uhh variable y which is a function of
x also So we can say that this is a functional
Similarly the area S of a surface founded
by a given curve is also a functional and
since this area is determined by the choice
of the surface So we have a curve c so let
us say that we have a curve c So this is say
the this is a curve now here Uhh this curve
A is Uhh bounded so we have a surface so we
can say we have a surface like this and any
you can consider like that
So we have a surface like this and this is
C is the curve bounds this surface in this
area So here we can we want to find out the
surface area of these Uhh surfaces which are
bounded by the curve c is it okay and we can
say that if we choose this surface then surface
area is different If we choose this surface
then the surface area is different and similarly
whatever Uhh surface you choose your surface
area is quite different
So we can say that here your surface area
S which is given by sat d of s over this reason
we can call it say D dash So we can find out
the surface area based on this D of s where
D of s is the say surface patch on this surface
D of s So if I take Uhh say 1 surface z equal
to z of xy right And then this can be written
as double integral over d and this is Uhh
the surface Uhh area element can be written
as 1 plus y dash means dy by d of x whole
square plusdz by d of x Whole Square and d
of x
And here D is the projection of this Uhh on
xy plane so here D is denoted as the projection
of your surface bounded by this scene on the
xy plane so Uhh sorry here it is dxdy So here
we can say that the (sur) area S of a surface
bounded by a given curve c is also a functional
Uhh because if we use different different
surface your surface area is different and
since this area is determined by the choice
of surface z which is given as z of xy
So so area S is given by this so here youre
functional related to surface area is given
by D integral over D under root 1 plus z as
square plus zy square double z by double y
square dx to ay Oh! So there is a small problem
here here it is Uhh here it is so there is
a wrong thing so here it is dz by dx whole
square plus dz by dy whole square okay Im
just correcting it
So here we can say that the surface area corresponding
to Uhh the surface Szxy is given by this double
integral D and the surface element can be
written as under root 1 plus double z by double
x whole square plus double z by double y whole
square dx dy and here is this D is the projection
of the area bounded by the curve c on the
xy plane so this is also an example of a functional
Now we may also define a functional Uhh in
the sense of centre of mass so if we represent
that this function y equal to yx is made of
some kind of material then we try to find
out the centre of mass of this material So
Uhh this y of yx will give you some kind of
a curve which is made by some material and
we try to find out say centre of mass here
So we can Uhh we have already seen that this
can be written as x bar equal to ratio of
2 functional A to B x under root 1 plus y
dash square d of x divided by A to B under
root 1 plus y dash square d of x
So Uhh this is the excomponent of Uhh the
centre of mass similarly you can find out
the y component of the centre of mass So in
in that case your y bar is defined by you
simply replaced by x by y So these this is
also the functional so it means that if you
change the Uhh y of x then your centre of
mass is going to change
So it means that if you change this curve
y it means if you are changing the shape of
the Uhh the shape of the material Uhh shape
of the Uhh say graph then its centre of mass
will also change So next thing is the that
the problem of minimal surface of revolution
this we have discussed that if we have a curve
y equal to y of x and is rotated about the
X axis through an angle to buy then the resulting
surface bounded by the plane x equal to a
and x equal to be as this area
So this we can write it like this this we
have in fact drawn but let me write it here
so here we have this is x this is y and this
is the Uhh graph y equal to y of x and this
is your A and this is B and this is rotated
Uhh through x axis of say okay So it is this
thing so now we try to find out say surface
area of this so if you look at here this we
can patch it these 2 entire (surf) entire
shape can be truncated into this kind of patch
and we can call this as patch as this is the
Uhh surface patch like d of s and we can say
that y is the say radius of this particular
batch and you can say that the surface area
is given by 2 pi y and d of s right
Now here Uhh since this is only line integral
so we are considering only say single so I
we can write it here 2 pi y d of s d of x
d of x So and this is we are considering only
between A to B so we can write it A to B So
we can say that the surface area of this Uhh
generated shape is given by A to B 2 pi y
ds by dx d of x So you can find out say d
of ds of dx as here you can write d of dsf
ds by dx is equal to under root 1 plus y dash
square
So here y dash represents d y by d of x so
here you can say that the Uhh resulting surface
bounded by the plane x equal to A and x equal
to B has this area S equal to 2 pi a to b
y 1 plus y dash square d of x and we Uhh so
this is also one example of functional and
if you look at in general we can define say
Uhh general example of functional as this
with the help of a continuous function f which
is continuous in terms of its Uhh arguments
So it means that function f is continuous
with respect to x y and z then you can define
functional defined as this J of y is equal
to a to b f of x y of x y dash x d of x So
here if you look at Uhh then by taking particular
values of this f you can say that this first
problem that is the length of the plane curve
and the surface area and the Uhh this the
Uhh area Uhh surface area Uhh is also can
be considered as a particular case of this
and here we will consider one property that
is this kind of functional is having a localisation
property
What is a localisation property That here
we can Uhh truncate the region of y of x for
example this particular problem Here we consider
this y of y of x and then we can truncate
the reason between a to b into smaller parts
and for each smaller part for example we have
this a to say x1 and then x1 to x2 and and
so on So here we can divide this Uhh entire
range a to b into several sub patches and
we can find out say corresponding functional
and find out the value of its functional
So it means that you can say that Uhh for
example here we have J of y is equal to Uhh
a to b and f of x y y dash d of x So here
in for this particular problem here f x y
y dash is given by 2 pi y dsx Uhh ds by dx
Now Im saying that if we want to write it
say J1y what is J1y Here you write it a to
x1 and f of x y y dash d of x So you can (si)
say that J1 is nothing but this Uhh sorry
So J1 is between a to x1 similarly you can
define J2 y is equal to your x1 to x2 f of
x y y dash d of x So similarly you can Uhh
find out Uhh patches find out say functional
defined in this sub patches and then if you
sum all these values then you will get the
values defined by this original linear functional
Uhh original function defined by JY equal
to a to b fx y y dash d of x so we can say
that this has a localisation property
But if we consider this part the functional
defined here this is not having this property
here we cannot do so we we call these kind
of function as nonlocal functionals and in
calculus of variations we generally deal with
the Uhh functional Uhh with the localisation
property So in calculus of variations we generally
deal with Uhh local functional not this norlocal
function
There are certain Uhh functions available
so methods available to deal this kind of
products they are for more general examples
we can consider a continuous function fx y
z which is a continuous function with respect
to its argument then a function may be defined
as J of y equal to a to b f of x y of x y
dash x d of x and here by taking a particular
examples of f consider that the problem this
problem which is the arc length of a plane
curve and the surface area of the surface
bounded by say Uhh a given curve c and the
problem of minimal surface of revolution Uhh
given by this equation number 2 is a particular
case of this right
So here Uhh summarizing all this we can say
that the calculus of variations is a branch
of calculus of functional is in which we find
the maximum and minimum of the functionals
and we can say that thus the calculus of variations
in this branch we discuss the methods that
permits Uhh finding maximal and minimal values
of functions and the problems which we investigate
R functional for a maximum or minimum are
called variational problems
In fact there are certain principle rules
in mechanics in physics that can be related
to Uhh functional that can be related as as
the maximising or minimising some kind of
functional in that process and they can that
that rules can be converted as a maximising
and minimising of that functional so that
kind of problems are also known as that kind
of rules are also known as variation principle
So that also we will discuss Uhh in calculus
of variations so for todays class Uhh we will
wind up things here In next class we will
discuss the concepts and basic basic definitions
of calculus of variation thank you for listening
us Thank you
