# Thread: Calculus of variations help needed

1. ## Calculus of variations help needed

I'm attempting to derive the maximum volume enclosed by a given surface area in a volume of revolution. I know the answer is a sphere!

As this is isoperimetric, I used lagrange multipliers for

integral pi.y^2.dx

integral 2.pi.y.ds

The ds must naturally be expressed as sqrt(1 + y').dx
as the previous function was expressed in terms of dx

L = pi.y^2 + 2.pi.y.sqrt(1+y')

and have attempted to put this through the Euler-lagrange equation. The solution is hesitant to be forthcoming, despite pages of manipulation.

1) There is a constraint on the start and end points of y(x) for the volume to be enclosed, namely y(x)=0 at the endpoints.
2) people have mentioned describing the relationship between the area under y(x) and the volume of revolution, but I can't force this into a helpful form!
3) as per usual L(x,y') would be more enticing than L(y,y') although this seems improbable.

The other question is to minimise the surface area for a given volume in a volume of revolution problem, but I assume that there will be a similarity in the two answers.

I have seen so much unhelpful hinting about how to get there...
A key piece of information or a full answer would be very, very much appreciated!

2. Originally Posted by mdruett
I'm attempting to derive the maximum volume enclosed by a given surface area in a volume of revolution. I know the answer is a sphere!

As this is isoperimetric, I used lagrange multipliers for

integral pi.y^2.dx

integral 2.pi.y.ds

The ds must naturally be expressed as sqrt(1 + y').dx
as the previous function was expressed in terms of dx

L = pi.y^2 + 2.pi.y.sqrt(1+y')

and have attempted to put this through the Euler-lagrange equation. The solution is hesitant to be forthcoming, despite pages of manipulation.

1) There is a constraint on the start and end points of y(x) for the volume to be enclosed, namely y(x)=0 at the endpoints.
2) people have mentioned describing the relationship between the area under y(x) and the volume of revolution, but I can't force this into a helpful form!
3) as per usual L(x,y') would be more enticing than L(y,y') although this seems improbable.

The other question is to minimise the surface area for a given volume in a volume of revolution problem, but I assume that there will be a similarity in the two answers.

I have seen so much unhelpful hinting about how to get there...
A key piece of information or a full answer would be very, very much appreciated!
A couple of things. First, your Lagrangian should be

$
L = \pi y^2 + 2 \pi \lambda y \sqrt{1+y'^2}
$
since $ds = \sqrt{1+y'^2}$ and not $\sqrt{1+y'}$

Second, since your Lagrangian is independent of $x$ the Euler-Lagrange equations can be written as

$
\frac{d}{dx} \left(\frac{\partial L}{\partial y'}\right) - \frac{\partial L}{\partial y} = 0 \;\;\; \Rightarrow\;\;\; \frac{d}{dx} \left( y' \frac{\partial L}{\partial y'} - L\right) + \underbrace{\frac{\partial L}{\partial x}}_{= \,0} = 0
$
so we have $y' \frac{\partial L}{\partial y'} - L = c$

With your $L$, and initial condition, the constant will be zero. Thus, you are left with a first ODE for $y$ to solve which is doable.

BWT - the sphere comes out!

3. The Lagrangian you suggested is the one I was using, so my fault for not checking what I'm typing!

I think you hit the nail on the head with the second suggestion.
Many, many thanks, Danny!
Will go and sort it out now...