What constraints can be imposed to a variational problem to render a surface with a given curvature?

Question

The problem was motivated by the following situation: Suppose we had a grinding device like the mortar with pestle as shown in the following picture

The pictured pestle may suggest a surface with either egg cross-section like

$$\tag{1} x^2+\frac{y^2}{\left(a+\frac{y}{b}\right)^2},\; 1.1\leq a\leq 1.5,\; 4\leq b\leq 10 $$ which is one solution of the ordinary differential equation (ode) $$\tag{2} y'=-\frac{x(ab+y)^3}{ab^3y}$$ Or I guess any cross-section among the circular, elliptical, parabolic or hyperbolic cross-section; respectively, with their corresponding ode's $$\tag{3} \begin{align} x^2+y^2=r^2&\to y'=\frac{x}{y}\\ \frac{x^2}{b^2}+\frac{y^2}{a^2}=0&\to xyy''+x(y')^2-yy'=0\\ (x-0)^2=4a(y-b)&\to xy''-y=0\\ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1&\to xyy''+x(y')^2-yy'=0\\ \end{align} $$

Instead of choosing any of those cross-sections, I wanted to obtain the "best grinding surface", from first principles. The largest assumption being made is that grinding is given by the largest grinding area, and for the moment physical considerations such as pressure or possibly angular momentum are not included.

The following are other assumptions being made:

1. Let $\mathcal{A}$ be the surface area of the pestle, with symmetry around the $z$ axis, and hence the description of the surface can be made through a cross section that can be parametrized through $\boldsymbol{\gamma}(t)=(x(t),y(t))$ or alternatively the cross section is given by the revolution around the $z$ axis of the curve $\boldsymbol{\gamma}(x,y(x))$
2. Assume that the "best shape" is given by the curve with the maximal curvature, and take the curvature $k$ to be defined as $$k=\frac{y''}{\left[1+(y')^2\right]^{3/2}}$$ if the curve is described by an explicit function $y=f(x)$.
3. Let the functional $L(x,y(x),y'(x)$, be defined as $L=k$.

Now, using the Euler-Lagrange equation $$\frac{\partial L}{\partial f}-\frac{d}{dx}\frac{\partial L}{\partial f'}=0$$

One obtains $$\tag{4} \frac{(-3y'y'')^{2/5}}{(1+(y')^2)}=c^{2/5}$$ or $$\tag{5} h(1+(y')^2)+(3y'y'')^{2/5}=0;\;h=c^{2/5}$$ Clearly, none of the cross-sections described with the ode's in (2) or (3) correspond to this result. Let us suppose that Eq.(5) is solved without making any assumptions. In Mathematica it renders 6 possible solutions. Just for illustration, one of them is $$\tag{6} y(x)\to \int_1^x -\sqrt{\frac{\sqrt[3]{-\left(h^2 K[1]+3 c_1\right){}^4}}{\sqrt[3]{h} \left(6 c_1 h^2 K[1]+h^4 K[1]^2+9 c_1^2\right)}-1} \, dK[1]+c_2$$ Here is the question I am intending to ask: Notwithstanding the proper annotations by John Huges, suppose we leave aside the grinding process itself for the moment and focuse on Eqs.(1-3); is there any way to solve Eq.(5), so that the result would be any of the described cross-sections at the beginning of this post (such as imposing boundary conditions to render something similar to Eqs. (1-3))? Am I making the correct assumptions? or are there some other geometrical considerations that may constrain even further the type of curvature of the discussed cross-sections in Eqs.(1-3)?

Answer 1

To be honest, I can't follow most of what you've written, but my Calculus of Variations knowledge is pretty lightweight. But I can comment on the modeling assumptions.

1. Surely the best grinding element's shape depends on the shape of the bowl. For instance, if the bowl is hemispherical, then the largest-area grinder is a ball that just fits into the hemispherical pocket.

2. The spherical example I just gave shows the limitation of the "largest area" as a target, but there's another limitation: if you take that grinder and carve lots of deep crevasses into it, you get even greater area, but it's useless area, because it's far from the opposing surface (and because grains of whatever you're grinding will be lost in the crevasses!).

3. Your item 2 says that the best shape has "maximal curvature", but surely the shape that achieves this is a knife-edge, whose curvature is infinite. On the other hand, in the "what I really want to ask" part, you veer away and say "maximal area", and at least for shapes that are smooth at some scale, those two goals may be in tension with one another.

In short: you might have an interesting differential-equations question, but it appears to me that it has nothing to do with the actual physical question from which you started.

In all this, the idea that "biggest area" is what matters is a huge error for all but the very smallest of mortars. Small contact area, and a tendency to not make the things being crushed "run away" from the compression point...THOSE are far more important design elements, along with manufacturability, strength, and ergonomics.