Physical Boundaries

Abstract

What Brâncusi intended was not to diminish the importance of shape and boundaries. He probably meant that there is more to a boundary than just the appearance of its shape (see Fig. 9.1). The shape can actually represent, through its freedom and its intangible majesty, the essence of reality. Shape is important, it is even essential, and sometimes, as Brâncusi said, it is hard to distinguish the reality it describes, as we can see in Fig. 9.2. Antoine de Saint Exupéry said (Citadelle 1948): “A rock pile ceases to be a rock pile the moment a single man contemplates it, bearing within him the image of a cathedral.” In other words, even though the content, the internal structure, and the density may be the same, what makes the difference here is the external shape.

When you see a fish you don’t think of its scales, do you? You think of its speed, its floating, flashing body seen through the water … If I made fins and eyes and scales, I would arrest its movement … I want just the flash of its spirit … What is real is not the external form, but the essence of things … it is impossible for anyone to express anything essentially real by imitating its exterior surface.

Constantin Brâncusi

Appendices

Appendix 1: Second Fundamental Form

Apart from the treatment of the geometry of graphs and networks, this book discusses surfaces embedded in $\mathbb{R}^{3}$, like the surface of a liquid drop. We thus present here some details of the differential geometry of smooth parameterized 2D surfaces embedded in $\mathbb{R}^{3}$. Moreover, when we discuss the expressions for the Hamiltonian or Lagrangian of free liquid surfaces, we use the concept of normal variation of a surface. In the following, we present a formal introduction to the variation of smooth parameterized compact surfaces embedded in $\mathbb{R}^{3}$. The discussion follows the work of Verpoort and Verstraelen [350]. The reader can supplement this section using various sources [136–140, 151, 152, 351, 352], where many more physical examples and abstract constructions can be found.

We consider a deformation of a compact surface $\varSigma \subset \mathbb{R}^{3}$ (in the following, we consider only smooth 2D parameterized surfaces embedded in $\mathbb{R}^{3}$) to be a smooth ‘deforming’ map $\mu: (-\epsilon,\epsilon )\times \varSigma \rightarrow \mathbb{R}^{3}$, $(t,p) \rightarrow \mu (t,p)$, $p \in \varSigma$, μ(0, p) = p, and $\varSigma _{t} =\mu (t,\varSigma )$. Let $\mathcal{X}(\varSigma )$ be the 3D vector bundle over $\varSigma$ and let ${\boldsymbol Z}_{p} \in \mathcal{X}(\varSigma )$ defined by

$$\displaystyle{{\boldsymbol Z}_{p} = \frac{\partial } {\partial t}\biggr |_{t=0}\mu (t,p)}$$

be the deformation vector field, i.e., the tangent vector describing the trajectories traced out by the point p when the surface begins to be deformed. We can generalize this operator to any geometrical object defined on $\varSigma$, and for any deformation field ${\boldsymbol Z}$. So we define the variation of a tensor T along the deformation vector field ${\boldsymbol Z}$ by

$$\displaystyle{\updelta _{{\boldsymbol Z}}T = \frac{\partial } {\partial t}\biggr |_{t=0}T\big(\mu (t,p)\big)\;.}$$

We define the shape operator S by its action on vector fields ${\boldsymbol V } \in \mathcal{X}(\varSigma )$:

$$\displaystyle{S: \mathcal{X}(\varSigma ) \rightarrow \mathcal{X}(\varSigma )\;,\qquad S({\boldsymbol V }) = -{\boldsymbol V }({\boldsymbol n}) = [{\boldsymbol n},{\boldsymbol V }]\;,}$$

where ${\boldsymbol n}$ is the inner unit normal to $\varSigma$, that is, the (negative) derivative of the unit normal in the ${\boldsymbol V }$ direction. Its eigenvalues are the principal curvatures and its eigenvectors are the principal directions in the surface. It is easy to check that the tangent variations of the mean and Gauss curvature are given by their Lie derivatives, i.e., if ${\boldsymbol V } \in T(\varSigma )$, then $\updelta _{{\boldsymbol V }}H ={\boldsymbol V }H$ and $\updelta _{{\boldsymbol V }}K ={\boldsymbol V }K$. Another interesting relationship in terms of S and H is $\nabla \cdot S = 2\nabla H$.

A deformation prescribed by the formula $\mu (t,p) = p + t{\boldsymbol Z}_{p}$ is called a linear deformation. Examples are translation of a plane or uniform expansion of a sphere. There is a very nice lemma [350] for the variation of the shape operator. Let ${\boldsymbol X} \in \mathcal{X}(\varSigma )$ and let ${\boldsymbol Z}$ be a linear deformation. Then,

$$\displaystyle{ \updelta _{{\boldsymbol Z}}S({\boldsymbol W}) = -{\boldsymbol W}(\updelta _{{\boldsymbol Z}}{\boldsymbol n}) - S({\boldsymbol W}){\boldsymbol Z} = [\updelta _{{\boldsymbol Z}}{\boldsymbol n},{\boldsymbol W}] +\big [{\boldsymbol Z},[{\boldsymbol n},{\boldsymbol W}]\big]\;. }$$

(9.92)

The variation of the mean curvature under a linear deformation is

$$\displaystyle{\updelta H = -\frac{1} {2}\bigtriangleup f + (K - 2H^{2})f +{\boldsymbol Z}^{t}(H)\;,}$$

where we make the decomposition ${\boldsymbol Z} = f{\boldsymbol n} +{\boldsymbol Z}^{t}$. As a corollary, we have a fundamental integral formula:

Theorem 14

Let $\varSigma$ be a compact surface, d ω the volume form, and ${\boldsymbol X}$ a smooth vector field defined on it. Then,

$$\displaystyle{\int _{\varSigma }\mathrm{div}{\boldsymbol X}\,\mathrm{d}\omega = 0\;.}$$

For a surface $\varSigma$ parameterized by the function ${\boldsymbol r}(u,v)$, we have the integral formulae of Jellett and Minkowski:

$$\displaystyle\begin{array}{rcl} \vert \varSigma \vert & =& -\int _{\varSigma }({\boldsymbol n},{\boldsymbol r})H\mathrm{d}\omega \;, {}\\ \int _{\varSigma }H\mathrm{d}\omega & =& -\int _{\varSigma }({\boldsymbol n},{\boldsymbol r})K\mathrm{d}\omega \;. {}\\ \end{array}$$

In [350], the author gathers four different geometric definitions for the second fundamental form on a surface, viz., $\varPi: \mathcal{X}(\varSigma ) \times \mathcal{X}(\varSigma ) \rightarrow C^{\infty }(\varSigma )$. We present these below and exemplify in Fig. 9.38.

Fig. 9.38

Four definitions of the second fundamental form of a surface. Upper left: Variation of ${\boldsymbol n}$ along a curve in dot product with a tangent vector. Upper right: Normal curvature κ _n of a curve Γ(t) lying on the surface M. The principal curvature of Γ is represented by κ. Bottom left: The second fundamental form also measures the distance between a point on the tangent plane and the surface. Bottom right: When a surface is deformed along its normal, the second fundamental form measures the rate of change of the area element

Definition 12

Let $\varSigma$ be a surface, Γ(t) a smooth curve parameterized by t lying on it, ${\boldsymbol V }$ the tangent vector to Γ at a point $p \in \varGamma \subset \varSigma$, and ${\boldsymbol W}$ another vector belonging to the tangent plane at p, ${\boldsymbol V },{\boldsymbol W} \in T_{p}(\varSigma )$. Then,

$$\displaystyle{\varPi ({\boldsymbol V },{\boldsymbol W}) = -{\biggl (\frac{\mathrm{d}} {\mathrm{d}t}{\boldsymbol n}(t),W\biggr )}\;,}$$

where ${\boldsymbol n}(t)$ is the inner unit normal to the surface along the curve Γ parameterized by t. This is illustrated in Fig. 9.38 (upper left).

Definition 13

Let Γ(t) be a parameterized curve lying on $\varSigma$, with ${\boldsymbol \tau }$ its tangent vector and κ _n its normal curvature on $\varSigma$. Then we have

$$\displaystyle{\kappa _{n} =\varPi ({\boldsymbol \tau },{\boldsymbol \tau }) \cdot {\boldsymbol n}\;.}$$

This is illustrated in Fig. 9.38 (upper right).

Definition 14

Consider a point $p \in \varSigma$ and a tangent vector to the surface at p, ${\boldsymbol V } \in T_{p}(\varSigma )$. Draw the line $s{\boldsymbol V }$ along this vector. Then,

$$\displaystyle{d(s{\boldsymbol V },\varSigma ) = -\frac{s^{2}} {2} \varPi ({\boldsymbol V },{\boldsymbol V }) + \mathcal{O}(s^{3})\;,}$$

where d is the distance from a point to the surface. This is illustrated in Fig. 9.38 (bottom left).

Definition 15

The most widely used definition is

$$\displaystyle{\updelta _{{\boldsymbol n}}I = -2\varPi \;.}$$

The second fundamental form is thus a measure of the variation of the first fundamental form (surface area element) along a normal deformation (see Fig. 9.38 bottom right).

As one can see, since the first fundamental form measures the element of area on the surface, whence it is an intrinsic characteristic of the surface (independent of its embedding), the second fundamental form is a measurement of how bent and twisted the surface is in some embedding in a higher-dimensional space, whether it be described by the behavior of its normal (Definitions 12 and 13), or by how much deviates from its tangent plane (Definition 14), or finally, by how much its area shrinks (Definition 15).

An infinitesimal deformation is an infinitesimal congruence if the displacement of any point of $\varSigma$ is along the normal at p to first order in the deformation parameter. An infinitesimal deformation is said to be isometric if the lengths of curves on the surface are stationary under the deformation. Then we have the Liebmann theorem which proves that any infinitesimal isometric deformation of the unit sphere is an infinitesimal congruence.

Appendix 2: Calculus of Variations

In the following V is a subset of the real vector space $\mathbb{R}^{n}$ with its canonical scalar product $\langle \;,\;\rangle$, and $\lambda \geq 0$ a real non-negative parameter. The set V is convex if for any two of its points x, y ∈ V the whole line segment from one to the other belongs to V, i.e., $\lambda x + (1-\lambda )y \in V$, $\forall \lambda \in [0,1]$. A function $f: V \rightarrow \mathbb{R}$ is a (strictly) convex function if

$$\displaystyle{f\big(\lambda x + (1-\lambda )y\big) \leq \lambda f(x) + (1-\lambda )f(y)\;,\quad \forall \lambda \in [0,1]\;,}$$

where for strict convexity we have strict inequality.

The function f has a global (local) minimum in V at x ₀ ∈ V if $\forall x \in V$ (in V intersected with some neighborhood of x ₀), f(x ₀) ≤ f(x), and we say that x ₀ defines a strict global (local) minimum, if the strict inequality holds $\forall x \in V$, x ≠ x ₀. If a strictly convex function on a convex set has a minimum, this minimum is unique on that set. Moreover, for the same function and $\forall x,y \in V$, we have

$$\displaystyle{f(y) - f(x) \geq \langle \nabla f,y - x\rangle \;,}$$

and

$$\displaystyle{\langle H(\,f)(x)y,y\rangle \geq 0\;,\quad y \in T(V,y)\;,}$$

that is, the function’s Hessian is semi-positive definite at all points belonging to the local tangent cone T(V, y) of V at y. By this cone, we understand the following construction (see Fig. 9.39). We define the cone of normals to V at y to be the closure of the set

$$\displaystyle{N(V,y) =\Big\{ z \in \mathbb{R}^{n}\vert \lambda z \in \nabla d_{ V }(y)\Big\}\;,}$$

for some $\lambda \in (0,\infty )$, where $d_{V }(y) =\min _{v\in V }\vert y - v\vert$ is the distance from the point y to the set V, and the gradient is calculated with respect to y. Then the local tangent cone T(V, y) of V at y is the orthogonal set to the cone of normals of V at y. Put loosely, the local tangent cone is the set of all half-lines starting from y which intersect the domain whose boundary contains V at least once. If x ₀ is a local minimum of f in V, then for any point z in the local tangent cone of V at x ₀, we have $\langle \nabla f,z\rangle \geq 0$, for all z ∈ T(V, x ₀). In addition, if f ∈ C ²(V ), we have

Fig. 9.39

The domain between the two straight cones is the local tangent cone for the parabolic cone presented in-between

$$\displaystyle{\nabla f = 0\;,\quad \langle H(\,f);z,z\rangle \geq 0\;,\;\;\mbox{ for all}\;z \in V \;,}$$

where H is the Hessian matrix of f viewed as a bilinear form. Actually, it can be proved that $\langle H(\,f);z,z\rangle /\langle z,z\rangle$ is the first (smallest) non-negative eigenvalue of the matrix H( f). Finally, we have the Kuhn–Tucker condition in the following form: if x ₀ is a local minimizer of f in V, and V is defined by a system of equations g _j (x) = 0, $j = 1,\ldots,m$ such that $\{\nabla g_{j}\}_{j=1,\ldots,m}$ is linearly independent, then there exists a set of parameters $0 \leq \lambda _{j} \in \mathbb{R}$ such that

$$\displaystyle{\nabla f +\sum _{ j=1}^{m}\lambda _{ j}\nabla g_{j} = 0\;,\qquad \sum _{j=1}^{m}\lambda _{ j}g_{j} = 0\;,}$$

and $L = f +\sum _{ j=1}^{m}\lambda _{j}g_{j}$ is called the Lagrange function.

In the following, we move from finite dimensions to variation of functionals. Let $u(x) \in B \subset C^{1}[a,b]$ be functions, $\|\cdot \|_{B}$ a certain norm on C ¹[a, b], and $E[u]: B \rightarrow \mathbb{R}$ a functional. The functional E is Gâteaux differentiable at u in direction h if there is a bounded linear operator $E'[u]: B \rightarrow \mathbb{R}$ defined by $E'[u] =\varPhi '_{u,h}(0)$, where Φ _u, h (t) = E[u + th]. We define the first and second Gâteaux variations of E at u in direction h ∈ B by

$$\displaystyle{ \updelta E[u](h) = \frac{\mathrm{d}} {\mathrm{d}t}E[u + th]\biggr |_{t=0}\;,\quad \updelta ^{2}E[u](h) = \frac{\mathrm{d}^{2}} {\mathrm{d}t^{2}}E[u + th]\biggr |_{t=0}\;. }$$

(9.93)

If E is Gâteaux differentiable, then we have

$$\displaystyle\begin{array}{rcl} \updelta E[u](h)& =& \langle E'[u],h\rangle \;,{}\end{array}$$

(9.94)

$$\displaystyle\begin{array}{rcl} \updelta ^{2}E[u](h)& =& \updelta \langle E'[u],h\rangle =\langle E''[u],h\rangle +\langle E'[u],h'\rangle \;.{}\end{array}$$

(9.95)

Furthermore, we say that E[u] is Fréchet differentiable at u ∈ B if there exists a bounded linear functional $DE: B \rightarrow \mathbb{R}$ such that

$$\displaystyle{E[u + h] = E[u] + DE[h] + \mathcal{O}(\|h\|_{B})\;,\quad \mbox{ as}\;\,\|h\|_{B} \rightarrow 0\;.}$$

If the functional is Fréchet differentiable then it is also Gâteaux differentiable, and E′[u] = DE[u].

If u ∈ B is a local minimizer for E in B, then $\langle E''[u]h,h\rangle \geq 0$ for all h in B. We now present the two main theorems of variational calculus.

Theorem 15

If $u \in \big\{ u \in B/u(a) = u_{1},u(b) = u_{2},u_{1,2} \in \mathbb{R}\ \mbox{ fixed}\big\}$ is a local minimizer of the functional

$$\displaystyle{E =\int _{ a}^{b}f(x,u,u')\,\mathrm{d}x}$$

in B, then u satisfies the Euler differential equation

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}x}f_{u'} = f_{u}\;.}$$

The theorem works equally for vector functions, i.e., $u: [a,b] \rightarrow \mathbb{R}^{n}$, whence the Euler equation becomes a system

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}x}f_{u'_{j}} = f_{u_{j}}\;,\quad j = 1,\ldots,m\;.}$$

Theorem 16

If u is a weak (strong) solution of the Euler equation (or system of equations), and if $\langle E''[u]h,h\rangle \geq 0$ for all h ∈ B, then u is a weak (strong) local minimizer of E in B.

By ‘weak’ or ‘strong’ in the theorem above, we mean satisfying the equation or the inequation in the weak sense, that is, in the C ¹[a, b] norm, or pointwise.

Appendix 3: n-Dimensional Rotating Drops

The stability of a rotating incompressible liquid drop, unaffected by gravity and cohered by surface tension, was the focus of an astonishing series of experimental and theoretical investigations starting with Joseph Plateau, the father of soap film studies, and pursued in the study of nuclear fission (the conjecture of Bohr and Wheeler in the model of heavy atomic nuclei), all the way to n-dimensional differential geometry generalizations.

In the following, we present a differential geometry model based on variational techniques and the implicit function theorem and which can be used to derive upper limits for the angular velocity as well as the existence, regularity, and stability of an energy minimizing family of rotating liquid drops in a neighborhood of the closed unit ball [353, 354].

We work in the n-dimensional Euclidean space $\mathbb{R}^{n}$ with coordinates x = (x ⁱ), and we define an incompressible liquid drop model as the class of compact and connected subsets of $E \subset \mathbb{R}^{n}$, n ≥ 2, with finite prescribed volume | E |  = Ω _n in the Hausdorff measure. Initially, the drop may have a unit sphere shape, that is | x |  _E (t = 0) = 1. We recall that the volume of the n-dimensional ball (disk) $D^{n} \subset \mathbb{R}^{n}$ in this measure is

$$\displaystyle{ \vert E\vert =\varOmega _{n} = \frac{\pi ^{n/2}} {\varGamma (n/2 + 1)}\;. }$$

(9.96)

In addition, we require the drop to have a class C ³ boundary ∂ E = M and to have its center of mass (or barycenter) placed at the origin and at rest:

$$\displaystyle{ \int _{E}x^{i}\,\mathrm{d}x = 0\;,\quad \forall i = 1,\ldots,n\;. }$$

(9.97)

For any positive angular speed Ω, we define the energy functional of the rotating drop model as the sum of the surface potential energy and the negative centrifugal kinetic energy:

$$\displaystyle{ \mathcal{F}_{\varOmega }(E) =\int _{M}\mathrm{d}A_{M} -\frac{\varOmega ^{2}} {2}\int _{E}\vert \pi _{\mathbb{R}^{n-1}}(x)\vert ^{2}\mathrm{d}x\;, }$$

(9.98)

where dA is the area form and $\pi _{\mathbb{R}^{n-1}}(x)$ is the orthogonal projection of the position vector centered at the center of mass onto the hyperplane x _n  = 0 perpendicular to the rotation direction. For star-shaped drops, the set E is a star-shaped region (that is, any straight line joining any point x in E to the origin belongs to E) and we chose a class $C^{3}(S^{n-1})$ map from the unit sphere parameterized by s to the boundary of E, viz., $X(s): S^{n-1} \rightarrow N \subset \mathbb{R}^{n}$. With this notation and | X |  = r(s), the drop surface is a compact Riemannian manifold with metric induced by the standard Euclidean metric on the sphere $g_{ij}^{0}(s)\,$:

$$\displaystyle{ g_{ij}(s) = r^{2}g_{ ij}^{0}(s) + \nabla _{ i}r\nabla _{j}r\;. }$$

(9.99)

The mean curvature of the drop surface is

$$\displaystyle{ H(r) = \frac{n - 1 -\vert \nabla ^{N}r\vert ^{2} - r\bigtriangleup ^{N}r} {r\sqrt{1 - \vert \nabla ^{N } r\vert ^{2}}} \;, }$$

(9.100)

where ∇^N and △^N are the tangent to the N surface gradient and the Laplace–Beltrami operator, respectively [136–138, 353, 354].

By calculating the critical points of the first variation of the energy under the constraints of volume preservation and constant position of the center of mass, we obtain the corresponding Euler–Lagrange equations for the rotating drop:

$$\displaystyle\begin{array}{rcl} H(r) -\frac{\varOmega ^{2}} {2}\vert \pi _{\mathbb{R}^{n-1}}(x)\vert ^{2}& =& X^{j}(s)\int _{ E}\left [H(r) -\frac{\varOmega ^{2}} {2}\vert \pi _{\mathbb{R}^{n-1}}(x)\vert ^{2}\right ]X^{i}\mathrm{d}x \\ & & \times \int _{N}X^{i}(s)X^{j}(s)\mathrm{d}A\;, {}\end{array}$$

(9.101)

where r = r(s), x ∈ E on the left-hand side, and the two integrals are taken in the sense of L ²(E) and L ²(N), respectively. The left-hand side represents the variation of the energy and the right-hand side the Lagrange multipliers of the two constraints. In [353, 354], it is proved that, for any ε > 0, (9.101) is solvable and has a smooth solution r _Ω (s) bounded by a closed neighborhood $\bar{B}_{\epsilon }$ of 0 for Ω < ε.

If r(s) is a solution for the rotating drop with angular velocity Ω, and if we find ε > 0 such that $\max _{s\in S^{n-1}}\vert r(s) - 1\vert \leq \epsilon$, there is a bounded function f(n) such that the solution is stable if

$$\displaystyle{ \varOmega < \frac{n + 1} {2} \big[1 -\epsilon f(n)\big]\;. }$$

(9.102)

The stability is understood in the sense of the solutions corresponding to stable energy minimizers of the energy (9.98) of the rotating drop. This theorem is proved using a Rayleigh quotient operator weakly identical to the second variation of the energy functional in the Fréchet derivative sense. A sketch of the dependence of the upper bound of the angular speed for stable solutions is presented in Fig. 9.40.

Fig. 9.40

Contour plot of the maximum angular speed of rotation Ω _max(n, ε) (in arbitrary units) for stable smooth solutions for the rotating liquid drop versus the number of dimensions of the space n, and the maximum deviation from the unit sphere of the drop shape. Smaller space dimensions and a more highly deformed shape decrease stability and the upper rotational limit

There is an interpretation of rotations in more than three dimensions if we consider the phase space of a system containing a large number of particles, like a statistical or thermodynamic system, where n approaches the value of Avogadro’s number. Consider a distribution function with compact and connected support in the phase space of a Hamiltonian dynamical system. According to the Liouville theorem, the motion of the density function through the phase space is a fluid flow of system points with zero convective derivative, that is, an incompressible flow [355]. In addition, the constant position constraint on the center of mass is satisfied simultaneously in both configuration and momentum space, so a system with its center of mass at rest will be represented by a phase space distribution whose center will also be at rest.

In principle, we can create the equivalent of a liquid drop in the phase space satisfying the constraints (9.96) and (9.97). We can induce a phase space rotation of the drop if this distribution is a Wigner distribution of a Gaussian wave packet in a quadratic trapping potential, like a harmonic oscillator. It is well known that the flow of the Wigner distribution follows circular paths in the phase space. It is also known that a straight motion in the phase space is equivalent to a Galilean transformation in the configuration space. This means that the states of a Galilean invariant many-body system constrained to have cyclic dynamics will perform liquid drop rotations in the phase space, with a property of inertia (tendency to conserve straight uniform translations in the phase space). In this context, we can assimilate the fluid motion in the phase space as the rotation of a liquid drop with constant volume, constant center of mass position, and a centrifugal field of forces.

Such a virtual motion can be examined formally by the above theoretical approach for n-dimensional drops in rotation. If we accept this model, we can infer from the stability criterion for the rotating drop shape presented above that an increase in the number of particles will allow rotations with higher angular speeds to remain stable. The more particles in a statistical system forced to oscillate, the higher the eigenfrequencies of its stable oscillations.

This observation may work like a universality principle for large amplitude high frequency oscillations in many-body systems: more particles and hence more collective interactions allow higher frequencies of stable oscillations. This observation is in agreement with many current models, theories, and experiments, like the Langumir waves (plasma oscillations in which the plasma frequency increases as the square root of the electron density), the oscillation frequencies of the collective excitations of a trapped Bose–Einstein condensate (the leading corrections to the frequencies are proportional to the number of atoms in the condensate to the power 1/5 [356]), the frequency of the collective oscillations of a trapped Fermi gas (the frequency is proportional to the square root of the electron concentration [357]), the self-maintained coherence effect in collective neutrino oscillations (the frequency increases with the neutrino gas density [358]), etc.