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Hi there. I am dealing with a mathematical problem which seems to be much harder than I initially expected:

Minimize the functional

[itex]J(\Omega) = \frac{1}{\rho} I_{z} = \int \!\! \int \!\! \int_\Omega \left( x^{2} + y^{2} \right) dx dy dz[/itex]

subject to

[itex] W(\Omega) = \frac{1}{\rho} x_c m(\Omega) = x_c \int \!\! \int \!\! \int_\Omega dx dy dz = \int \!\! \int \!\! \int_\Omega x dx dy dz = C = constant[/itex]

i.e.the unknown to be optimized for is the domain of integration [itex]\Omega[/itex]. How to solve this problem as generally as possible? Shall one assume that: a) [itex]\Omega[/itex] is continuous? b) [itex]\Omega[/itex] is differentiable, and (if yes) in which sense?

Those who are familiar with mechanics immediately notice that the problem in fact is: Assuming constant density [itex]\rho[/itex] throughout the body, minimize the Moment of Inertia [itex]I_{z}[/itex] around the z-axis while keeping the Moment [itex]x_c m[/itex] around the same axis constant. Anyway, the problem as it stands is of purely mathematical nature so I think it belongs to this section.

This is what I tried so far,

- Introducing a Lagrange multiplier [itex]\lambda[/itex]

[itex]\bar{J}(\Omega) = J(\Omega) + 2 \lambda W(\Omega) = \int \!\! \int \!\! \int_\Omega \left( x^{2} + y^{2} + 2 \lambda x \right) dx dy dz = \int \!\! \int \!\! \int_\Omega \left( (x + \lambda)^{2} + y^{2} - \lambda^{2} \right) dx dy dz[/itex]

- Deciding what [itex] \Omega[/itex] should look like

[itex]\bar{J}(\Omega) = \int_{z_l}^{z_u} \!\! \int \!\! \int_{\Omega_z} \left( (x + \lambda)^{2} + y^{2} - \lambda^{2} \right) dx dy dz = \int_{z_l}^{z_u} \!\! \int_{y_l(z)}^{y_u(z)} \!\! \int_{x_l(y,z)}^{x_u(y,z)} \left( (x + \lambda)^{2} + y^{2} - \lambda^{2} \right) dx dy dz[/itex]

- Deriving equations for the extremals of [itex]\bar{J}(\Omega)[/itex]

[itex] \bar{J}_{\! x_l} = - \int_{z_l}^{z_u} \!\! \int_{y_l(z)}^{y_u(z)} \left( (x_l(y,z) + \lambda)^{2} + y^{2} - \lambda^{2} \right) dy dz = 0[/itex]

[itex] \bar{J}_{\! x_u} = \int_{z_l}^{z_u} \!\! \int_{y_l(z)}^{y_u(z)} \left( (x_u(y,z) + \lambda)^{2} + y^{2} - \lambda^{2} \right) dy dz = 0[/itex]

[itex] \bar{J}_{\! y_l} = - \int_{z_l}^{z_u} \!\! \int_{x_l(y_l(z),z)}^{x_u(y_l(z),z)} \left( (x + \lambda)^{2} + y_l(z)^{2} - \lambda^{2} \right) dx dz = 0[/itex]

[itex] \bar{J}_{\! y_u} = \int_{z_l}^{z_u} \!\! \int_{x_l(y_u(z),z)}^{x_u(y_u(z),z)} \left( (x + \lambda)^{2} + y_u(z)^{2} - \lambda^{2} \right) dx dz = 0 [/itex]

[itex] \bar{J}_{\! z_l} = - \int_{y_l(z_l)}^{y_u(z_l)} \!\! \int_{x_l(y,z_l)}^{x_u(y,z_l)} \left( (x + \lambda)^{2} + y^{2} - \lambda^{2} \right) dx dy = 0[/itex]

[itex] \bar{J}_{\! z_u} = \int_{y_l(z_u)}^{y_u(z_u)} \!\! \int_{x_l(y,z_u)}^{x_u(y,z_u)} \left( (x + \lambda)^{2} + y^{2} - \lambda^{2} \right) dx dy = 0[/itex]

where [itex] \bar{J}_{\! \cdot} = \frac{\partial \bar{J}}{\partial \ \cdot} [/itex].

After this point I am kind of stuck. I am not even sure that the expressions in 3. are correct, but I believe so. I have not been able to fully evaluate any of the integrals in 3. or even analyze them in any other meaningful way. Obviously, by looking at the integrand of the triple-integral, a coordinate transformation along the x-axis is possible, followed by a transformation to polar coordinates. The integrand then becomes [itex] ( r^{2} - \lambda^{2}) r[/itex] which doesn't seem to be any simpler. After the transformation the domain of integration is still unknown so nothing has been gained.

HELP?

PS. I believe that problems like this one must have been solved ages ago, but I couldn't find anything. If anybody knows the solution to the problem and the proof thereof, please let me know. A conjecture from my side is that if the object is finite in the z-direction,i.e.[itex]z_{u}-z_{l}[/itex] is finite, then the optimal shape is a half-disk with thickness [itex]z_{u}-z_{l}[/itex] with obvious orientation. If the thickness of the disk is allowed to increase, it extends along the z-axis and becomes more and more "slender". When [itex]z_{u}-z_{l} \rightarrow \infty[/itex] the radius of this half-disk goes to zero. This conjecture that goes along well with intuition is what I am trying to prove, however.

EDIT: I think I had some of the equations for the extremals in 3. wrong. They are now corrected. I would be happy if someone would check them for me.

EDIT 2: I think I had the equations correct the first time. They are now again corrected! Please still check them for me.

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# Minimizing Moment of Inertia, keeping Moment constant

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