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Math Help - Calculus of variations help needed

  1. #1
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    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.

    I have had several ideas, but none have led to progress... please help!

    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!
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  2. #2
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    Quote Originally Posted by mdruett View Post
    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.

    I have had several ideas, but none have led to progress... please help!

    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

     <br />
L = \pi y^2 + 2 \pi \lambda y \sqrt{1+y'^2}<br />
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

     <br />
\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<br />
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!
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  3. #3
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    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...
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