Hadamard Formula

How does the shape of a violin's body influence its tone? How does a subtle change to an airplane's wing alter its lift? These questions touch upon a deep and fundamental relationship: the connection between an object's geometry and the physical laws that govern it. While intuition provides partial answers, mathematics offers a precise and elegant tool for this inquiry, known as the Hadamard formula. This formula addresses the critical challenge of quantifying exactly how a system described by a partial differential equation responds to a change in its physical domain.

This article serves as a guide to this powerful principle. In the first chapter, ​​Principles and Mechanisms​​, we will dissect the formula itself, exploring what each component means in the context of vibrating systems like strings and drums. We will see how it provides a universal law of boundaries, connecting motion, stress, and change. Subsequently, in ​​Applications and Interdisciplinary Connections​​, we will witness the formula in action, journeying through diverse fields to see how this single idea is used to calculate physical forces, understand quantum phenomena, design optimal structures, and even probe abstract concepts in pure mathematics. Let's begin by uncovering the inner workings of this remarkable key that unlocks the secrets between shape and physics.

Imagine you are a master luthier, crafting a fine violin. You know, with an artisan's intuition, that the precise shape of the instrument's body is paramount. A millimeter's difference in the curve of the wood can change the richness and color of its tone. Or picture an aerospace engineer designing a wing. The shape of the wing's cross-section determines its lift and drag, and even tiny modifications can have dramatic effects on performance. In both cases, a fundamental question arises: how, precisely, does a change in the shape of an object affect the physical phenomena that occur within and around it?

This is not just a question for luthiers and engineers. It lies at the heart of many fields of science. The answer, in many profound cases, is given by a remarkable piece of mathematics known as the ​​Hadamard formula​​. It is our master key to understanding the deep and often surprising relationship between geometry and the laws of physics, described by partial differential equations.

Let's think about a simple vibrating system, like a drumhead stretched over a frame. The sounds it can make correspond to a set of characteristic frequencies. In the language of mathematics, these frequencies are related to the ​​eigenvalues​​ of the Laplacian operator, and the corresponding modes of vibration are the ​​eigenfunctions​​. Let's call the fundamental (lowest) frequency's eigenvalue $\lambda_1$. It depends on the shape of the drum, which we'll call the domain $\Omega$. So, we write $\lambda_1(\Omega)$.

Now, suppose we slightly nudge the boundary of the drum. We take our domain $\Omega$ and deform it into a new domain $\Omega_t$ by moving each point on the boundary by a tiny amount. This movement is described by a "velocity field" $V$ on the boundary. The question is: what is the new eigenvalue $\lambda_1(\Omega_t)$?

The Hadamard formula provides a stunningly elegant answer for the rate of change of the eigenvalue as we begin the deformation:

This formula might look intimidating at first, but it is a poem written in the language of mathematics. Let's read it together, word by word.

At its core, the formula says that the change in the eigenvalue is an integral—a sum—of contributions from every point along the boundary $\partial\Omega$. What determines the contribution at each point? Two things: how the boundary moves, and how the system behaves there.

• ​​The Motion: $(V \cdot n)$​​ This term represents the component of the velocity field $V$ that is perpendicular (or ​​normal​​) to the boundary. It tells us how fast the boundary is moving straight outwards (if positive) or inwards (if negative) at a particular spot. If the boundary is just sliding along itself (tangentially), this term is zero. The formula tells us that, to first order, such tangential sliding has no effect on the eigenvalue. Only the motion that actually changes the domain's area or volume matters.

• ​​The Stress: $(\frac{\partial u}{\partial n})^2$​​ This is the most physically insightful part. The function $u$ is the eigenfunction, representing the shape of the vibration. For a drumhead, $u=0$ on the boundary because it's clamped down. The term $\frac{\partial u}{\partial n}$ is the ​​normal derivative​​ of $u$. It measures how steeply the drumhead lifts off from the clamped edge. You can think of it as a measure of the stress or tension at that point on the boundary. Where the vibration is steepest near the edge, the stress is highest. The formula includes the square of this term, which means two things: it's always positive, and it gives much greater weight to regions of high stress.

• ​​The Whole Picture​​ Putting it all together, the Hadamard formula tells us that the eigenvalue is most sensitive to shape changes in the regions where the stress is highest. If you want to lower the pitch of your drum most effectively, push the boundary outwards where the drumhead is vibrating most vigorously near the edge.

  And what about the minus sign? It encodes a deep physical principle. If we expand the domain (so $V \cdot n > 0$ on average), the integral will be positive. The minus sign tells us that $\frac{d\lambda}{dt}$ will be negative. In other words, ​​making a domain bigger lowers its fundamental frequency​​. This is exactly what we expect from experience! A longer guitar string has a lower pitch; a bigger drum has a deeper boom. The Hadamard formula doesn't just confirm this intuition; it quantifies it exactly.

The true power of a physical principle is revealed in its application. Let's see how the Hadamard formula works its magic on a few classic shapes.

• ​​The Vibrating String​​ The simplest vibrating system is a 1D "domain"—a string of length $L$. Its fundamental eigenvalue is $\lambda_1(L) = (\frac{\pi}{L})^2$. If we stretch the string by a tiny amount, the formula tells us precisely how the eigenvalue changes. As a direct consequence, we can calculate how the eigenvalue $\mu_1 = 1/\lambda_1$ of the inverse operator changes. It turns out that $\frac{d\mu_1}{dL} = \frac{2L}{\pi^2}$, a beautifully simple relationship showing that this quantity grows quadratically with length.

• ​​The Perfect Drum: A Circular Disk​​ Let's move to a 2D drumhead in the shape of a unit disk. What happens if we expand it uniformly, like blowing up a circular balloon? Here, the velocity is constant, $V \cdot n = c$. The fundamental vibration mode $u$ on a disk is radially symmetric, meaning the "stress" term $(\frac{\partial u}{\partial n})^2$ is the same at every point on the boundary. The integral becomes easy, and the formula gives a wonderfully compact result: the rate of change is simply $-2c\lambda_1$. The relative change in eigenvalue is twice the relative change in radius!

  But what if we apply a more complex, wavy perturbation, like $\cos(2\theta)$, which pushes some parts of the boundary out and pulls others in? The "stress" term is still constant around the rim. However, the perturbation term $\cos(2\theta)$ averages to zero over the circle. The pushes and pulls perfectly cancel each other out, and the formula predicts a change of... zero! To first order, the fundamental frequency doesn't even notice this kind of sloshing deformation. This highlights the crucial role of symmetry.

• ​​The Humble Square​​ Domains don't have to be perfectly round. Consider a unit square. Its fundamental vibration mode is a gentle mound, like $u_0(x, y) = 2 \sin(\pi x) \sin(\pi y)$, which is zero on the boundaries. The "stress" is highest in the middle of each side and drops to zero at the corners. If we expand the square by pulling it outwards from the origin with a vector field $V(x,y) = (x,y)$, the Hadamard formula allows us to compute the change by integrating along the four edges. The two edges along the axes don't contribute because the velocity field is zero or tangential there. The change is driven entirely by the expansion of the far edges at $x=1$ and $y=1$, leading to the crisp result $\lambda'_1(0)=-4\pi^2$.

The genius of the Hadamard formula is that its principle extends far beyond eigenvalues. It is a general statement about how quantities defined by PDEs change with their domains.

• ​​The Energy of a Field​​ Consider a charged object held at a certain electric potential. The total electrostatic energy stored in the field is given by the ​​Dirichlet energy​​, an integral of the squared gradient of the potential field. How does this energy change if we deform the object? Once again, a version of the Hadamard formula provides the answer. It involves a boundary integral that depends on the boundary's motion, the potential, and its derivatives there. This has direct applications in fields like capacitance and heat flow.

• ​​The Green's Function: A System's Response​​ In physics, the ​​Green's function​​ $G(z, w)$ is a profoundly important tool. It represents the response of a system at point $z$ to a single, sharp "poke" or unit source at point $w$. For a membrane, it's the shape the membrane takes when you poke it at one point. The Hadamard formula tells us how this response function itself changes when we wiggle the boundary. By integrating the product of the normal derivatives of two Green's functions (one for the source, one for the observation point) against the boundary's motion, we can find the change in this fundamental solution.

From vibrating strings and drumheads to the energy of electric fields and the fundamental response of a physical system, the Hadamard formula offers a single, unifying perspective. It reveals that the global properties of a system are intimately tied to the local behavior at its boundary. The change is always an integral over the boundary, a weighted sum of the boundary's normal motion. The weighting factor is always a measure of the "activity" or "stress" of the physical field at that point.

This is a recurring theme in physics and mathematics: the secrets of the interior are often written on the boundary. The Hadamard formula is one of the most beautiful and powerful expressions of this deep truth, a testament to the elegant interplay between the shape of space and the laws that unfold within it.

We have spent some time understanding the machinery of shape derivatives and the Hadamard formula. It is a beautiful piece of mathematics, certainly. But what is it for? What good is knowing how some abstract functional changes when we wiggle the boundary of a domain? It turns out this is not an idle question for mathematicians. It is a question that Nature asks, and answers, everywhere. The response of a physical system to a change in its geometry is a fundamental aspect of our world, and the Hadamard formula is our key to understanding it. Let us take a journey through a few of these worlds and see this principle at work.

Perhaps the most direct and intuitive application of these ideas is in calculating forces. A force, after all, is related to the change in energy with respect to a change in position. What if that "position" is not the location of a particle, but the location of a boundary?

Imagine a metal conductor, charged with electricity. It sits in space, surrounded by an electric field. This field contains energy. We know from our previous discussions that the electric field will arrange itself to minimize this energy for the given geometry. Now, suppose we slightly bulge a tiny patch of the conductor's surface outward. The shape has changed, so the field must readjust to find a new minimum-energy configuration. The total energy in the field will change. By the principle of virtual work, this change in field energy must equal the work done by the electrostatic force pushing the surface outward. The Hadamard formula gives us a direct way to calculate this energy change due to the boundary deformation. It tells us that the change is proportional to the energy density right at the surface. By equating this to the work done, $P \times dA \times \delta n$, we can solve for the pressure, $P$. We find, with remarkable simplicity, that the outward pressure on the conductor is proportional to the square of the electric field normal to the surface. The charged conductor is not static; it is alive with an outward pressure, a desire to expand, which we can precisely quantify by "wiggling" its boundary.

This idea is not confined to the macroscopic world of conductors. It echoes in the strange realm of quantum mechanics. Consider a single electron trapped in a two-dimensional "box" with impenetrable walls. Its existence is described by a wavefunction, and it can only possess certain discrete energy levels. What if one wall of the box is not perfectly flat but has some microscopic roughness? We can think of this roughness as a small outward bulge on the boundary. This changes the geometry of the box. How does this affect the electron's energy?

Just as with the conductor, we can use the Hadamard formula. The energy of the electron is an eigenvalue of the Schrödinger equation. The formula tells us that the change in this energy is given by an integral over the boundary of the deformation multiplied by the square of the normal derivative of the wavefunction. This derivative represents the "quantum pressure" the electron's wavefunction exerts on the wall. For an outward bulge, which increases the volume of the box, the energy change is negative. The particle has more room to roam, so its confinement energy decreases. It's a beautiful parallel: the classical electrostatic pressure and the quantum mechanical energy shift both spring from the same fundamental principle of how energy responds to a change in shape.

In 1966, the mathematician Mark Kac asked a famous question: "Can one hear the shape of a drum?" What he meant was, if you know all the resonant frequencies (the eigenvalues) of a vibrating membrane, can you uniquely determine its shape? While the general answer turned out to be no, the question launched a field called spectral geometry, which explores the deep connection between the geometry of a domain and the spectrum of operators like the Laplacian.

The Hadamard formula is a central tool in this field. It answers a related question: "If I slightly change the shape of a drum, how do its frequencies change?" The frequencies of a drum correspond to the eigenvalues of the Laplacian with Dirichlet boundary conditions. The formula tells us that the first-order change in any eigenvalue is an integral over the boundary, involving the normal displacement and the square of the normal derivative of the corresponding eigenfunction.

What is fascinating is that some deformations might not change a frequency at all, at least to first order. If we perturb the boundary of a circular drum with a sinusoidal wiggle like $\cos(m\theta)$, the change in the fundamental frequency is exactly zero. This is because the eigenfunction for the fundamental tone is perfectly symmetric, and the wiggles of the perturbation average out to nothing when integrated against it. The system's response depends not just on the perturbation, but on the state it is in.

This connection between geometry and spectrum has a wonderful probabilistic interpretation. The lowest eigenvalue, $\lambda_1$, governs the long-term behavior of a system. Imagine releasing a drop of ink—a random walker undergoing Brownian motion—inside our domain. The ink spreads out, but is removed the instant it touches the boundary. The survival probability of the ink at long times decays exponentially, as $\exp(-\lambda_1 t)$. A smaller $\lambda_1$ means a slower decay and a longer survival time.

When we deform the domain, we change its principal eigenvalue $\lambda_1$. The Hadamard formula tells us exactly how. An outward deformation ($v_n \ge 0$) makes the domain larger. Intuitively, it should take longer for the random walker to find the boundary. This means the survival probability should increase, and the decay rate $\lambda_1$ should decrease. The formula confirms this perfectly: the change in $\lambda_1$ is proportional to $-\int (\partial_n \phi_1)^2 v_n \, d\sigma$, which is negative for an outward deformation. Thus, the Hadamard formula beautifully connects the geometric act of enlarging a domain to the probabilistic fate of a random walker within it.

We can even go further and ask about the properties of the entire spectrum at once by packaging it into a single object, the spectral zeta function, $\zeta_D(s) = \sum_n \lambda_n^{-s}$. This function is of profound importance in quantum field theory and number theory. And yet, the Hadamard formula can be extended to tell us how this complex object varies as we change the shape of the domain $D$.

So far, we have been analyzing how properties change with a given deformation. Let's turn the tables. What if we want to design a shape to achieve a specific goal?

This is the domain of topology optimization, a revolutionary tool in modern engineering. Suppose you want to design a bridge or an airplane wing. You have a certain amount of material, and you want to distribute it to create the stiffest possible structure. "Stiffness" means minimizing the deformation (compliance) under a given load.

The compliance is an energy functional that depends on the shape of the material domain. The Hadamard formula allows us to compute the "shape derivative" of this functional. This derivative is a function defined on the boundary of our current design, and it tells us something remarkable: at each point on the surface, it tells us how much the overall stiffness would improve if we added or removed an infinitesimal bit of material there. It is a gradient, but not in space—it is a gradient in the infinite-dimensional space of all possible shapes.

By following the direction of steepest descent—that is, by removing material where the shape derivative is positive and adding it where it's negative—we can iteratively evolve our design towards an optimal shape. Starting with a simple block of material, this process carves out intricate, often organic-looking structures that are perfectly tailored to their function. It is a powerful demonstration of the Hadamard formula not just as an analytical tool, but as a creative one.

Finally, let us see how this one idea blossoms in the more abstract fields of pure mathematics, where its elegance and unifying power are perhaps most apparent. In potential theory and complex analysis, mathematicians study concepts like "capacity," which are subtle measures of the size of a set.

The Newtonian capacity in three dimensions or the logarithmic capacity in two dimensions can be defined as the minimum energy of a certain potential function. Since they are defined by minimizing an energy integral, their sensitivity to domain perturbations is naturally described by the Hadamard formula. The change in capacity is given by an integral over the boundary involving the square of the normal derivative of the equilibrium potential.

Here too, we find surprising simplicities. If we take a circular annulus and wiggle its outer boundary, the first-order change in its conformal modulus—a key geometric invariant—depends only on the average change in radius. The higher-frequency wiggles all average out to zero. In another beautiful example, calculating the change in capacitance for an ellipse under a rather complicated-looking boundary perturbation reveals that the complexities in the perturbation formula and the underlying conformal map cancel each other perfectly, yielding an astonishingly simple result.

The formula even reaches into the heart of complex analysis, connecting the shape of a domain to the zeros of analytic functions within it. Jensen's formula relates the positions of a function's zeros to the average of its logarithm on the boundary. The boundary term is an integral, and the formula for the Green's function, which depends on the domain, is subject to Hadamard variation. By combining these, one can calculate how the "potential" of the zeros changes as the boundary is perturbed. This reveals a deep and unexpected link between the geometry of the boundary and the analytic structure of functions inside.

From the palpable pressure on a conductor to the esoteric structure of analytic functions, the Hadamard formula provides a common thread. It shows us that in field after field, the response of a system to a change in its boundaries is not arbitrary, but follows a universal and elegant law—a law that relates the global change in a quantity to a local integral over the boundary where the action happens. It is a testament to the profound unity of mathematical and physical principles.