February’s topics:

Math: a bridge between the “two cultures.”

Gordon Gillespie maintains that mathematics can help “bridge the yawning gulf” between the humanities and the sciences.

In a recent article in Aeon, the Australian magazine of ideas, Gillespie starts by sketching out the landscape of intellectual thought. In his perspective it has two main communities, the humanities and the sciences, “The Two Cultures” as C. P. Snow characterized them in his 1959 book. Gillespie writes: “Off the record, most natural scientists still consider the humanities to be a pseudo-science that lacks elementary epistemic standards,” and “Meanwhile, many humanities scholars see scientists as pedantic surveyors of nature, who may produce practical and useful results, but are blind to the truly deep insights about the workings of the (cultural) world.”

Gillespie sets up two extreme archetypes. For the naive realists, phenomena can only be explained by a causal chain back to mechanistic first principles. They believe that “nature is ordered according to laws that operate regardless of whether or not humans are around to observe.” The naive idealists believe the mind and the social and cultural apparatus it has generated over the millenia evolve in a separate sphere with laws of its own. They “insist that all order is conceptual order, which is based solely on individual or collective thought.” It is certainly true that mathematics occupies a middle ground between these extreme positions: 17 will always be a prime but much of the rest, although constrained by logic, is a cultural construct. I believe, for example, that the mathematics developed on a planet orbiting some star in the Andromeda nebula would make perfect sense to us once we studied it, but that the concepts evolved there would likely be completely different from ours.

Gillespie writes for an audience conversant in Kant and Wittgenstein, but he finds some nice specific examples to buttress his argument that modern science’s use of mathematics forces it to depart from the narrow doctrine of causality. One is the twin paradox in relativity, in which one young twin spends a couple of years touring the universe at near the speed of light and comes back to find that his brother is an old man. As Gillespie explains, there is no “physical cause” that determines their age difference: “the true reason lies in the structural framework” in which the phenomenon occurs: the geometry of space-time. This geometry may be mathematical in structure, but it has become part of modern physics. Another example he gives involves the Central Limit Theorem. Causality can explain exactly how a penny can spin and land, but there is no purely causal explanation for the bell-shaped distribution of outcomes if a thousand pennies are tossed. Nevertheless the normal distribution, a completely human construct, is an indispensable part of the scientific toolbox.

The essay ends with the reflection that all the sciences are “inseparably rooted” both in the world outside us and in the human mind, and that this duality, uniquely obvious in mathematics, resides beyond causality.

Möbius strips in water waves.

Within the past several years, researchers began theoretically investigating how topological structures like vortices and Möbius strips might form in water waves. A new paper in Nature describes an experimental apparatus to make these phenomena happen. The authors are an international team of eight led by Bo Wang (Fudan University and Henan University) and Zhiyuan Che (Fudan).

In the proposed experiment, the key to topological phenomena is the angular-momentum vector field of the waves. Angular momentum is the rotational analogue of forward momentum; it’s what keeps a spinning top spinning. It is not obvious that water waves carry angular momentum, but they do. An ideal particle floating on the surface undergoes a circular “orbital” motion as the wave passes: it is lifted, pushed forward, dropped and pulled back. There’s a nice animation on YouTube illustrating this movement. The particle motion traces out a circle in a vertical plane oriented in the direction of propagation of the wave, with diameter approximately equal to the height of the wave. If the particle has mass, there is an angular momentum associated to its circular motion. Angular momentum is represented by a vector perpendicular to the circle, pointing as a corkscrew would move if turned in that direction. That direction, independent of the mass, is represented by the vector field $S$ in this illustration.

A sinusoidal water wave propagating left to right in the x-direction, with amplitude in the z-direction, and constant in the y-direction. In the troughs lies a particle which propagates along a clockwise-oriented circle.
A schematic illustration of a plane wave (here moving from left to right), the circular motion of two floating particles, and the corresponding angular momentum vectors. The particles (yellow dots) rotate clockwise in a vertical plane; the red angular momentum vector field $S$ is parallel to the $y$-axis. Image credit: Tony Phillips.

To make use of this angular momentum, the authors built a hexagonal pool with 16cm sides. Sides 1, 3 and 5 are driven by a sinusoidal signal to produce water waves with wavelength about 4cm. Sides 2, 4 and 6, are stationary. This generates three waves which travel at angles of 120$^{\circ}$ from each other. The driving signals are also adjusted so the waves are out of phase with each other, also by 120$^{\circ}$.

A schematic view of the apparatus: a. Three of the sides of the hexagonal pool are sinusoidally excited to generate waves.
b. The three sets of waves have the same frequency and amplitude, but are offset 120$^{\circ}$ in direction and in phase. Image credit: Tony Phillips.

When instead of a single wave we have three waves interfering, the motion of particles on the surface becomes more complicated. The following image shows the corresponding $S$-field. The vectors are color-coded according to their direction, with the color intensity signaling their amplitude.

Graphic showing regions of (moving clockwise): Pink, yellow, blue, pink, yellow, blue. Near the center the graphic is studded with triangles where the colors meet, interspersed with black triangles (regions of low amplitude).
The angular momentum vector field $S$ resulting from the superposition of three plane waves with the same frequency and amplitude, but offset by 120$^{\circ}$ in direction and in phase. Amplitude and phase of the $S$-vectors are represented by intensity and hue of the color-coding. Image courtesy of Bo Wang.

With three waves, the excitation pattern varies from point to point. Instead of always describing circles, the particles can move in ellipses or be stationary. At a point where the motion is a non-circular ellipse, one can draw a line along the major axis of the ellipse. The authors point out how, around a singularity of the angular-momentum field $S$, those lines can trace out a Möbius strip.

Top: Zoomed-in image of the center of the previous image. Within the enlarged square, we can analyze the field of spin directions. The coordinates give positions in the surface of the pool. Bottom: Following the dashed circle around a singularity of the $S$-field, the principal axes of the ellipses traced out by the wave displacement vectors form a Möbius band. Because in this image those axes are shown as oriented, at some point in the circle around the singularity the indicated orientations must flip. These points are highlighted in yellow. Image courtesy of Bo Wang, partially edited.

The authors go on to show that in a set-up with 24 interfering water-waves instead of 3, the interference patterns can be controlled so as to move floating particles from one place to another. This gives their paper its title: “Topological water-wave structures manipulating particles.”

Some additional information is in a press release posted on Miragenews.com.

Emmy Noether, symmetry and conserved charge.

On February 4, New Scientist posted “The 100-year-old symmetry theorem that is still changing physics today,” an article about mathematician Emmy Noether. Emmy Noether is revered by topologists for unleashing the full power of algebra into what became the field of algebraic topology. I recommend Friedrich Hirzebruch’s account of her influence.

The authors here (John Gribbin and Mary Gribbin) focus on another achievement, simply known as “Noether’s Theorem,” which continues to be fundamental in theoretical physics. As they describe it, “It was Noether who proved that energy must be conserved—in any theory—if the laws of physics stay the same no matter what time it is—in other words, if they are time-invariant.” They go on to elaborate that her theorem does not only apply to time-invariance: Whenever a physical system has a continuous symmetry, Noether’s theorem says there must be a corresponding conserved quantity. For example, in quantum mechanics, only phase differences matter. If all the phases in a quantum-mechanical system are moved forward or back by the same angle, the physics is exactly the same. This is a symmetry whose corresponding conserved quantity is electrical charge.

Noether’s Theorem applies to systems where the dynamics can be encapsulated in a function called a Lagrangian (or, equivalently, in a Hamiltonian). The Lagrangian was introduced by Joseph Louis Lagrange (1736-1813) as a purely mathematical convenience. It makes it significantly easier to derive the equations of motion for a complicated mechanical system.

The Lagrangian for a system is a function of the relevant positions, which we can write as a vector ${\bf x}$, the velocities ${\bf v}$, and time. The usual notation is ${\cal L}({\bf x}, {\bf v}, t)$. The idea was that the system would evolve between times $t_0$ and $t_1$ in such a way as to minimize the integral of ${\cal L}$. This integral was called the action but has no direct physical significance. In general, for a function like ${\cal L}$ to be the solution to an integral-minimization problem like this one, it has to satisfy a differential equation known as the Euler-Lagrange equation (this is part of the calculus of variations). For a mechanical system, the Euler-Lagrange equation gives the equations of motion (see below).

The way Noether’s Theorem works can be seen in the example of a mass $m$ on a frictionless horizontal table tethered by a spring to a wall. When the spring is stretched or compressed from its rest position, it exerts a restoring force $F=-kx$ proportional to the deformation $x$ and pointing in the opposite direction. For a 1-dimensional mechanical system like this one, the Lagrangian is

$${\cal L}(x, v, t) = \frac{1}{2}mv^2 – \frac{1}{2}kx^2.$$

Here, the quantity $\frac{1}{2}mv^2$ is the kinetic energy of the mass, and $\frac{1}{2}kx^2$ is the potential energy: the work it takes to stretch (or compress) the spring from its rest position to $x$.

A simple mechanical system: a mass $m$ sliding on a frictionless table, coupled to a spring with spring constant $k$. When the spring is stretched (or compressed) a distance $x$ from its resting position, it pulls (or pushes) back with a force proportional to $x$. Image credit: Tony Phillips.

This Lagrangian has time-invariance in a very specific sense. If the clock is advanced by an infinitesimal amount $\epsilon$, then $x$ is infinitesimally updated to $x+\epsilon \frac{dx}{dt} = x + \epsilon v$, and $v$ is updated to $v+\epsilon \frac{dv}{dt}$, which we write as $v+\epsilon {\dot v}$. The new Lagrangian ${\cal L}’$ is then $\frac{1}{2}m(v~+~\epsilon{\dot v})^2 ~-~ \frac{1}{2}k (x~+~\epsilon v)^2.$ Neglecting higher powers of $\epsilon$, this is

$$\frac{1}{2}mv^2~ +m\epsilon v {\dot v}~-~\frac{1}{2}kx^2 ~-~ k\epsilon xv = {\cal L}(x,v,t)+\epsilon(mv{\dot v}~-~kxv).$$

The change is $\epsilon(mv{\dot v}-kxv)$, which is $\epsilon$ times the time-derivative of the function $K(t)$ defined by $K(t) = \frac{1}{2}mv^2 -\frac{1}{2}kx^2$. This means that if we integrate ${\cal L}’$ from $t_0$ to $t_1$ to calculate the new action, we will get

$$ \int_{t_0}^{t_1} {\cal L}'(x,v,t) dt = \int_{t_0}^{t_1} {\cal L}(x,v,t) dt + \epsilon\int_{t_0}^{t_1} \frac{dK}{dt} dt.$$

By the Fundamental Theorem of Calculus, this becomes

$$ \int_{t_0}^{t_1} {\cal L}'(x,v,t) dt = \int_{t_0}^{t_1} {\cal L}(x,v,t) dt + \epsilon(K(t_1) – K(t_0)).$$

Since they differ by a constant, minimizing one integral is the same as minimizing the other: The physics has not changed. In general, in the context of Noether’s Theorem, invariance with respect to a coordinate $s$ means that an infinitesimal change in $s$ changes the Laplacian by the time-derivative of some function (and therefore leads to the same minimum of the action). This case is especially simple because the parameter $s$ is time itself.

Noether’s Theorem applied to this case states that since the Lagrangian is time-invariant, the quantity $v\frac{\partial{\cal L}}{\partial v}-K$ must be conserved. In this case $\frac{\partial{\cal L}}{\partial v}=mv$, so that quantity is $mv^2 ~-~ K = mv^2 ~-~ (\frac{1}{2}mv^2 ~-~\frac{1}{2}kx^2) = \frac{1}{2}mv^2 ~+~\frac{1}{2}kx^2$, the sum of the kinetic and potential energies: If the Lagrangian is time-invariant, the total energy of the system must be conserved.

One last note: the Euler-Lagrange equation for a problem like this is $$\frac{\partial L}{\partial x} -\frac{d}{dt}\frac{\partial L}{\partial{v}}=0.$$
Since in our example $\frac{\partial L}{\partial x}=-kx$ and $\frac{\partial L}{\partial v}=mv$, the Euler-Lagrange equation is $-kx-m\frac{dv}{dt}=0$ or $-kx=m\frac{d^2x}{dt^2}$, the familiar Newtonian $F=ma$ for a mass coupled to a spring.

Thanks to my colleague Martin Roček for help with this item.

—Tony Phillips, Stony Brook University

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