But the cookie dough riding the belt observes the circular stamp moving toward it and length *contracted* along the belt. How can the cutter *contracted* along the belt cut cookies *stretched* along the belt?

*The cut is simultaneous in the factory frame but not simultaneous in the cookie dough frame!* Clocks synchronized in their own frame are not synchronized in a moving frame, and the rear clock is ahead of the front clock. In the looping animation, the top panel illustrates the cut in the factory frame, where the cutter moves up & down simultaneously, while the bottom panel illustrates the cut in the dough frame, where the cutter rear moves up & down first and the cutter front moves up & down later. The wave of up-&-down motion spreads the cut over time lengthening it.

The relativistic effects of** time dilation**, **length contraction**, **clock desynchronization**, and the **relativity of rigidity** are significant only near the billion-kilometers-per-hour invariant speed of light and gravitational waves.

I understood how **Maxwell’s equations** describe **electromagnetism **long before I understood polarized outlets. The key is that north American homes tap transformers in three places to get a “neutral” and *two* “hots” that alternate but oppositely!

Starting at the top of the figure, a power plant wire loop rotates in a magnetic field inducing AC according to **Faraday’s law of induction **(one of Maxwell’s equations). The first transformer increases the voltage and lowers the current for energy efficient long distance transmission (again as described by Faraday). The second transformer decreases the voltage and increases the current for safe residential use. In north American homes, which use **split-phase power**, three wires enter the home: a neutral wire from the center of the final transformer coil, which is zeroed or **grounded**, and two hot wires from the ends of the coil, where current oscillates sinusoidally but 180° out of phase. Connecting a lamp across a hot and neutral wire harnesses a **root-mean-square** 120 volts of energy per charge, while connecting an oven across the two hots harnesses RMS 240 V. The polarized outlet forces appliances to connect their loads to neutral and their switches to one of the hots, so no current flows across their loads when the appliances are off, which is both safe and efficient.

So begins an article I co-authored with **NAIL**, the Nonlinear Artificial Intelligence Lab at **NCSU**, which appears today in **Physical Review E**.

Inspired by how brains work, artificial **neural networks** are powerful computational tools. Natural neurons exchange electrical impulses according to the strengths of their connections. Artificial neural networks mimic this behavior by adjusting numerical **weights and biases** during training sessions to minimize the difference between their actual and desired outputs.

From cancer diagnoses to self-driving cars to game playing, neural networks are revolutionizing our world. But although they are **universal approximators**, their approximations may require exponentially many neurons. In particular, they can be confounded by the mix of **order and chaos** in natural and artificial phenomena.

NAIL’s solution to this “chaos blindness” exploits an elegant and deep structure to everyday movement discovered by **William Rowan Hamilton**, who remarkably re-imagined Isaac Newton’s laws of motion as an **incompressible energy-conserving flow** in an abstract, higher-dimensional space of positions *and* momenta.

In this **phase space**, any motion is a unique trajectory confined to a constant-energy surface, and regular motion is further confined to a donut-like **hypertorus**. This structure constrains our Hamiltonian neural networks to properly forecast systems that mix order and chaos.

Like the **Tokyo Olympics**, the 2020 **pandemic** has postponed — but not cancelled — this summer’s NSF REU program until next summer.

**STATIONARY ACTION**

For simple systems, the Lagrangian is the difference between the kinetic and potential energies,

L = T - V.Demand that the action

S = \int dt \,Lbe stationary and integrate by parts

0 = \delta S = \int dt \left(\frac{\partial L}{\partial x} \delta x + \frac{\partial L}{\partial \dot x} \delta \dot x \right) = \int dt \left(\frac{\partial L}{\partial x}- \frac{d}{d t}\frac{\partial L}{\partial \dot x} \right)\delta x,for all variations \delta x, to find the Euler-Lagrange motion equation

\frac{\partial L}{\partial x} = \frac{d}{d t}\frac{\partial L}{\partial \dot x}.**CLASSIC OSCILLATOR**

For the simple harmonic oscillator, the Lagrangian

L[x,\dot x] = \frac{1}{2} m \dot x^2 - \frac{1}{2} k x^2implies the Euler-Lagrange equation

-kx = m \ddot x,which recovers familiar laws of Hooke and Newton.

**SPINLESS PARTICLE**

In 1+1 dimensional spacetime, represent a spinless particle by the quantum of the field \phi[t,x]. In units where c = 1 and \hbar = 1, the Lagrangian

L = \int dx\, \mathcal L = \int dx \left( \mathcal{T} - \mathcal{V} \right)and the Lagrangian density

\mathcal{L}[\phi, \partial_t \phi, \partial_x \phi] = \frac{1}{2} \left(\partial_t \phi \right)^2 -\frac{1}{2} \left( \partial_x \phi \right)^2 -\frac{1}{2} m^2 \phi^2,where the signs of the kinetic energy terms reflect the signs of the spacetime interval d\tau^2 = dt^2 - dx^2. The corresponding Euler-Lagrange equation

\frac{\partial \mathcal{L}}{\partial \phi} = \frac{\partial}{\partial t}\frac{\partial \mathcal{L}}{\partial (\partial_t\phi)} + \frac{\partial}{\partial x}\frac{\partial \mathcal{L}}{\partial (\partial_x\phi)}reduces to the Klein-Gordon equation

-m^2 \phi = \partial_t^2 \phi - \partial_x^2 \phior

\left(\partial_t^2 - \partial_x^2 + m^2 \right) \phi = 0.Assuming wave-particle duality, seek plane wave solutions

\phi = e^{i(-Et+ px)} = e^{i(-Et+ px)/\hbar}= e^{i(-\omega t+ kx)}to find

\left( (-iE)^2 - (i p)^2 + m^2 \right) \phi = 0or

E^2 = p^2 + m^2,where m is the particle mass (and E = m = m c^2 for p = 0).

Nonzero mass determines the simple harmonic oscillator potential energy \mathcal V = m^2 \phi^2 / 2 > 0 and the curvature of the corresponding upright parabola. Zero mass collapses the Klein-Gordon equation to the electromagnetic wave equation.

**SYMMETRY BREAKING**

Consider the massless Lagrangian density

\mathcal{L}= \frac{1}{2} \left(\partial_t \phi \right)^\dagger \left(\partial_t \phi \right) -\frac{1}{2} \left( \partial_x \phi \right)^\dagger \left( \partial_x \phi \right) + \frac{a}{2} \phi^\dagger \phi - \frac{b}{4} \left( \phi^\dagger \phi \right)^2,where \phi = \phi_m e^{\phi_a} = \phi_r + i \phi_i , the adjoint \phi^\dagger = \phi^* reduces to complex conjugation, and the parameters a,b > 0. For all complex arguments \phi_a, the potential energy density

\mathcal{V} = - \frac{a}{2} \phi_m^2 + \frac{b}{4} \phi_m^4depends only on the complex magnitude \phi_m. The vanishing derivative

0 = \frac{d \mathcal{V}}{d \phi_m} = - a \phi_m + b \phi_m^3= \phi_m(- a + b \phi_m^2)implies local maximum and minimum at \phi_m = 0 and \phi_m = \sqrt{a/b} = \phi_0, as in the figure, where the circular symmetry reflects the invariance of the Lagrangian under global phase (or gauge) transformations \phi \rightarrow \phi \, e^{i \Lambda} = \phi_m e^{i( \phi_a + \Lambda )}.

If the potential is a sombrero, the circular brim are ground or vacua states of nonzero field

\phi_r^2 + \phi_i^2 = \phi_0^2.Break the circular symmetry by fixing \phi_r = \phi_0 and \phi_i = 0, but allow the field to oscillate radially and shift circularly

\phi[t, x] = \phi_0 + \phi_1[t, x] + i \phi_2[t, x] .The Lagrangian density becomes

\begin{array}{c}\displaystyle\mathcal{L}= \frac{1}{2} \left(\partial_t \phi_1 \right)^2 -\frac{1}{2} \left( \partial_x \phi_1 \right)^2 - a \phi_1^2 \\ \displaystyle+ \frac{1}{2} \left(\partial_t \phi_2 \right)^2 -\frac{1}{2} \left( \partial_x \phi_2 \right)^2 \\ \displaystyle- \sqrt{a b}\,\phi_1 \left(\phi_1 ^2+\phi_2 ^2\right) -\frac{b}{4} \left(\phi_1 ^2+\phi_2^2\right)^2\\ \displaystyle+ \frac{a^2}{4 b}\end{array}or

\begin{array}{c}\displaystyle\mathcal{L}= \frac{1}{2} \left(\partial_t \phi_1 \right)^2 -\frac{1}{2} \left( \partial_x \phi_1 \right)^2 - m_1^2 \phi_1^2 \\ \displaystyle+ \frac{1}{2} \left(\partial_t \phi_2 \right)^2 -\frac{1}{2} \left( \partial_x \phi_2 \right)^2 - m_2^2 \phi_2^2 \\ \displaystyle+\text{interaction} + \text{constant},\end{array}where m_1 = \sqrt{a} > 0 and m_2 = 0.

**SUMMARY**

Starting with a massless field \phi with nonzero vacuum states, symmetry breaking creates a field \phi_1, corresponding to parabolic radial motion and a massive quantum m_1 > 0, and a field \phi_2, corresponding to constant circular motion and a massless quantum m_2 = 0 (a Goldstone boson). This is the Higgs mechanism of mass endowment.

]]>It was a great pleasure and honor to talk to them.

]]>Today, over 400 operational satellites occupy **geosynchronous** or **Clarke** orbit, peering at Earth from almost a tenth of the way to Luna. And this week for the first time, a servicing spacecraft, Northrop Grumman’s **Mission Extension Vehicle-1** (MEV-1), rendezvoused and docked with one of them, **Intelsat-901**, which was nearly out of station-keeping fuel. The capture mechanism went through the throat of the Intelsat-901’s **apogee engine**, which was not designed for docking. MEV-1 will use its **ion engines** to orient the stack and extend Intelsat-901’s operation for another 5 years. MEV-1 will then move Intelsat-901 to a slightly higher** graveyard orbit **— before rendezvousing and docking with *another* satellite to extend *its* lifetime.

The conference takes place in a small village in the Harz, a Mittelgebirge (I didn’t know that this is an English word!) in Northern Germany.

Because I had time, I chose to walk/hike the 7 miles from the train station in the next larger town Goslar to the hotel. It was wonderful and my most relaxing travel to a conference site ever. Because of the rain in the last days, some parts where pretty wet and some streams got larger than usual. But other people ‘built’ already crossings.

One thing I remembered, after it was too late: A shortcut is not always the best path to take. If you save on distance traveled in a mountainous regions you could pay with an increase in slope! Not surprising but interesting to realize.

After my talk, listening to presentations, and talking to many friends and colleagues, I hiked back on Tuesday. I stopped at a lake and a bear cave until the sun started to get pretty low. This is when I took a picture of my gigantic shadow.

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