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I am having difficulty formulating a problem, which involves optimizing a contour shape, into a well-posed variational form that would give a reasonable answer.

Within a bounded region on the $xy$ plane, say $x\in[-x_{0},x_{0}], y\in[-y_{0},y_{0}]$, we have a continuous scalar field $H=H(x,y)$. Both the field and the geometry of the problem exhibit no variations in the $z$ direction (i.e. $\partial/\partial z=0)$. On the plane within the specified region, there exists a closed loop (contour) $g(x,y)=0$ that encloses and defines a planar area $A$ that is penetrated by the field $H$. It is known from the physics of the problem that varying the shape of the contour $g$ can result in extremising the functional

$$ J(y):=\iint_{A} H(x,y)dA $$

and I need to find the optimal shape $g$ of the contour, for a given $H$ function that is nontrivial ($\neq 0)$ and captured area $A$ that is nonzero. In writing $J$ here I assumed that $x$ is the independent variable and $y$ is dependent on it, to draw the contour shape.

Anticipating that the closed contour function will be most likely expressible in parameteric form $(x(t),y(t))$, and since classical variational formulations I am familiar with usually deal with paths rather than areas, I tried to write the functional in terms of the contour (instead of the area) as follows, using Green's theorem:

$$ J=\iint_{A} H(x,y)dA=\iint_{A} \left(\frac{\partial F_{y}}{\partial x} -\frac{\partial F_{x}}{\partial y}\right)dA=\oint_{g}(F_{x}dx+F_{y}dy) =\int_{t=0}^{2\pi}(F_{x}\dot{x}+F_{y}\dot{y})dt $$

where $\boldsymbol{F}=F_{x}\hat{i}+F_{y}\hat{j}$ is some vector field whose curl may be defined to give $H$ (assuming we can find such field), dotted symbols like $\dot{x}$ denoting derivate in parameter $t\in[0,2\pi]$, and $\oint_{g}$ denoting integral around closed contour $g$. So, we can think of the Lagrangian of this problem as $L(x,y,\dot{x},\dot{y}):=F_{x}\dot{x}+F_{y}\dot{y}$.

The problem now is, if I don't impose any further contraints, the two Euler-Lagrange equation here (in $t$ now as independent variable and both $x$ and $y$ as dependents) give the same result (instead of two independent answers), which says that $H=0$. I plugged in different test fields $H$, and this is always the answer.

If I try to improve the formulation by imposing a constraint that $\iint_{A}dA=A_{0}$ to make the area nonzero constant, thus:

$$\frac{1}{2}\int_{0}^{2\pi}(x\dot{y}-y\dot{x})dt=A_{0} \Rightarrow \int_{0}^{2\pi}\left[ \frac{x\dot{y}}{2}-\frac{y\dot{x}}{2}-\frac{A_{0}}{2\pi} \right]dt=0,$$

giving a new (constrained) Lagrangian as $L(x,y,\dot{x},\dot{y},\lambda):=F_{x}\dot{x}+F_{y}\dot{y}+\frac{\lambda}{2}(x\dot{y}-y\dot{x}-\frac{A_{0}}{2\pi}),$

then, again, the two Euler-Lagrange equations in $x$ and $y$ give the same answer, basically that $H=\lambda$, where $\lambda$ is Lagrange's multiplier for this contraint.

What is wrong with my formulation, and how do I make it well posed for this problem so I can proceed?

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  • $\begingroup$ What is the "physics of the problem"? $\endgroup$ – Keith McClary Sep 15 '17 at 4:51
  • $\begingroup$ @KeithMcClary [1/2] The problem represents a case of resonance, where two types of energies become equal at resonance, and they are (each) defined as integrals: $\iint_{A_{1}}W_{1}dA$ and $\iint_{A_{1}}W_{2}dA$, where $A_{1}$ is the initial domain. So, resonance will occur when their difference is zero, hence the functional $\iint_{A_{1}}WdA$, where $W=W_{1}-W_{2}$. We know already the solution of $W$ to this problem of domain $A_{1}$. The issue now comes when this field is faced with a new geometric feature inserted in the domain, perturbing it slightly but not sufficiently to change ... $\endgroup$ – user135626 Sep 15 '17 at 23:17
  • $\begingroup$ @KeithMcClary [2/3] ... its eigenfunctions, and hence $W$ is assumed to be unchanged. The inserted feature is a "hole" of region $A\subset A_{1}$ and thus the new domain is now $A_{1}-A$, giving new functional as $\iint_{A_{1}-A}WdA=\iint_{A_{1}}WdA-\iint_{A}WdA=0-\iint_{A}WdA$. It is observed by physical experiment that, although W (related to the resonant eigenfunctions) is approximatly fixed, the eignevalues (wavelength) will shift, subject to choice of SHAPE of the contour enclosing this "hole", which may be tweaked to reach a resonant point same as the original wavelength (eigenvalue)... $\endgroup$ – user135626 Sep 15 '17 at 23:24
  • $\begingroup$ @KeithMcClary [3/3] ... even if the area $A$ is fixed (say $A_{0}$). We are thus seeking the optimal shape of the contour enclosing this "hole" region, which will restore the functional to zero. Clearly, this contour shape will reflect the way the present fields $W_{1}, W_{2}$ (or their difference $W$) vary in space. So, I tried to attack the problem as I wrote in the question above, hoping to reach an answer. I also tried plugging in example values for $W_{1},W_{2}$, such as the simple case of $B \sin k x$ and $C \cos k x$, but no reasonable progress is reached, as explained above. $\endgroup$ – user135626 Sep 15 '17 at 23:31
  • $\begingroup$ What exactly do you mean by the extremum of a vector-valued function? (Or is $H$ actually real-valued despite you call it a "field"?). From what you wrote in comments, it seems like you actually have a scalar function $H$ and just want a domain over which the integral is $0$ but then you start talking about "eigenfunctions that do not change" and that makes me totally confused... $\endgroup$ – fedja Sep 18 '17 at 8:12

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