Welcome, we are trying to understand a variation
calculus so we will start with differentiation
what is differential suppose.
If you have a function right y as a function
of X so that is some curve now when we take
derivative what we do is that we take a small
change in X and find out what is the corresponding
change in Y and dy/dx is basically said to
be the differential right this is the process
of differentiating on the other hand when
we are looking at variation. What we are looking
at is that we are looking at the same x value
but we are looking at the curves which have
different slope at the same x value and different
Y at the same x value so basically we are
looking at for example there could be another
thing which goes like that so at the same
x value it has a different Y value and it
has a different slope so you can also consider
another function which is like.
That which means that the same x value it
has a different Y value and it has a different
dy/dx one so these are known as variations
so in other words when we are looking at a
variation okay no change in X so are going
to look at the same x value at the same x
value we are looking at functions which have
different y + dy/dx value at that point so
that is why y+ dy/dx become the independent
variables and we are trying to look.
At the what happens to these things so at
this point where for the same X the Y and
the dy/dx are not changing are said to have
no variation right there is 0 variation on
at these two points okay those are our X1
and X2 points in our definition yesterday.
So there is away to make it a more formal
so like we say dy/dx the d/dx which is the
differential so similarly we define what is
known as a delta operator and delta operator.
Operating on I for example is nothing but
the varied I minus the original I now we define
this Δ operator in such a way that it obeys
certain properties and one of the properties
is that if you take Δdy then it is the same
as d of ΔY that is the variation on the differential
is the same as the differential on the variation
similarly we also define Δ acting on Ydx
integral is nothing but integral of ∂ydx
that is the variation on the integral is nothing.
But integral on the variation so if these
sort of properties are fulfilled then we can
define what is known as a delta operator and
the sink the derivation that we are going
to dousing the Δ operator is the same as
what we did in the last lecture for obtaining
the variation except that we are making the
process more formal and hence it becomes you
know probably you have done it when you did
the derivative initially.
The derivative is defined as a limit some
hedge tending to zero F(x)+H-F(x)/H and every
time you want to take the derivative a given
function then you have to evaluate this quantity
and then you have to find out what the derivative
is but after a while then we come up with
formula like d/dx of sinx is this or d/dx
of x squared is this and soon and so forth.
So it becomes more mechanical in a similar
fashion it is possible to make the process
of taking variational derivative a little
bit more formal which is what I want to do
so.
The derivation proceeds identical to what
we did in the last class except that the notation
and the terminology we use is a little bit
different from what we did so in that process
so let us take the original functional that
we considered yesterday and try to do the
variational derivative.
So what is the function will be considered
we considered I is equal to integral f of
x y’dx okay so we want to consider the so
we want to define as the variation on yΔy
is nothing but Y tilda minus y okay, y tilda
are the varied pass remember we had the one
optimizing path which was identified as Y
and all the varied paths about that we're
basically defined by Y tilde so y tilde minus
y is Δy so Δy’.
And now because we also have all this commutation
between Δ and D so also has this property
so it is y tilda prime minus y’ okay. So
we now look at F so let us take fxy + Δy’+
Δy’ what is this quantity. So we are going
to tell it expand like earlier so it is xy’+
∂/∂ y Δy + ∂/∂y’Δy’ plus higher
order terms in Δ. Now if we take this so
f of xy + Δyy’ + Δyy’ minus this f of
x, y, y’b are going to call this as the
total variation on earth. So now what is this
total variation on F by the expansion that
we have written.
We can see that Delta total on F is nothing
but 0 f/yyΔy+∂/∂y’Δy’ plus order
of Δ2 now I am also going to define what
is known as the Δ total acting on I which
is nothing but the, so we have to integrate
between x1 to x2. And so we had this yes so
X0y+Δy+Δy’ right, so this is what we are
going to call the total Δ so that is nothing
but Δ so this quantity if I call as Δ1 of
f plus order of Δ2.
So this quantity delta total of I is nothing
but an integral x1 to x2 which is Δ1 of I
okay, so what is the Δ1 of I so the first
variation on I is nothing but X 1 to X 2 ∂/∂y
Δy+ ∂/∂y’Δy’ integrated over dx
okay. So that Δ total of I is nothing but
Δ1 of I plus order Δ2 +Δ1 of I we identified
with this Δ1 of because this is tilted total
of F. So this is Δ1 of f order Δ2 so Δ1
of I will be this integral now we can do now
the integration by parts on this quantity.
So we have Delta one of I should be equal
to zero which is a necessary condition so
that means X 1 to X 2 ∂/∂yΔy- you know
Δ and D are interchangeable. So when we so
let me write the first full quantity and then
we will do that Δy’dx. So which I am going
to write as integral x1 to x 2 ∂/∂Y Δy
I am going to write outside and ∂f/∂y’Δy’
and Δ are interchangeable.
So I am going to make it as d by dx(Δy) and
that d/dx I am going to bring on to this side
by integration by parts. So that gives me
minus d/dx(∂f/∂y’dx) and this integration
by parts we do we also have an extra surface
term which is nothing but ∂ if by ∂y’Δy
evaluated at points x1 and x2 and the variation
ΔY is 0 at the x1 and x2.
So we are going to take this quantity off
remember it need not be 0 you still have ∂f/∂y’
to be 0 on x1 or x2 is sufficient we will
look at that case with respect to the action
principle that we derived just now at the
end of this derivation. So if this one t becomes
zero then we know that this should be equal
to zero because the first variation should
be equal to 0 and the same argument as earlier
works because this Δy is an arbitrary variation
in Y and this total integral from this x1
to x2 should be zero.
Then within that domain this should be identically
equal to zero which is basically the Euler
Lagrange equation that we have derived
So by so we have this ∂f/∂y – d/dx of
low f by ∂y’ should be equal to zero and
that is the so-called Euler Lagrange equation
this is what we derived and we derived this
in the last lecture so we have done the same
thing except that we have used this so called
Δ operator formalism. Now let us go back
to the term that we made 0 on the surface,
so that term was ∂f by ∂y’Δy at X1
and X2 let us consider this quantity with
respect to the action principle that we were
looking at in the case of action what is this
quantity so this quantity was something like
this and this was Δx right.
So at these p1 t 2 so this Δx then is the
variation in X variation in position so this
is if you make this equal to 0 that is equivalent
to prescribing the position at time t1 and
t2 so this is or prescribing the position
or displacement boundary condition so you
can prescribe them then in which case there
is no variation allowed at time t1 and t2
at these points so they will become 0 on the
other and you do not have to prescribe the
position you can expect this ∂ f by∂.
X dot to be prescribed and if you look at
that quantity that is nothing but the force
at that point okay so you can also describe
the force at any given point you can give
both at the point so either you can prescribe
a force or you can prescribe a displacement
at a given point in either case of prescribing
them makes this term on the surface to go
to zero so you get the Euler Lagrange equation.
So this becomes very important there is a
nice text book by shames and dim called the
energy and the finite element methods thirds
in structural mechanics I think so which basically
deals with the boundary conditions that one
can naturally derive from the variational
calculus okay so in any case so if you assume
that your variations are prescribed at the
endpoints you can assume this quantity to
be zero so you get the Euler Lagrange equation
which is what we derived in the last lecture
also so this is the variation derivative.
So basically variation derivative or the Euler
Lagrange equation is the first variation on
the functional that we are considering so
this brings us to the question as to why in
the case of modeling we need to deal with
variations why is it that we are dealing with
functional and not with functions so that
is what we will discuss in the next part of
this lecture. Thank you.
