An Efficiency Greater Than 1?

No, my next project is not building a Perpetuum Mobile.

Sometimes I mull upon definitions of performance indicators. It seems straight-forward that the efficiency of a wood log or oil burner is smaller than 1 – if combustion is not perfect you will never be able to turn the caloric value into heat, due to various losses and incomplete combustion.

Our solar panels have an ‘efficiency’ or power ratio of about 16,5%. So 16.5% of solar energy are converted to electrical energy which does not seem a lot. However, that number is meaningless without adding economic context as solar energy is free. Higher efficiency would allow for much smaller panels. If efficiency were only 1% and panels were incredibly cheap and I had ample roof spaces I might not care though.

The coefficient of performance of a heat pump is 4-5 which sometimes leaves you with this weird feeling of using odd definitions. Electrical power is ‘multiplied’ by a factor always greater than one. Is that based on crackpottery?

Heat pump.

Our heat pump. (5 connections: 2x heat source – brine, 3x heating water hot water / heating water supply, joint return).

Actually, we are cheating here when considering the ‘input’ – in contrast to the way we view photovoltaic panels: If 1 kW of electrical power is magically converted to 4 kW of heating power, the remaining 3 kW are provided by a cold or lukewarm heat source. Since those are (economically) free, they don’t count. But you might still wonder, why the number is so much higher than 1.

My favorite answer:

There is an absolute minimum temperature, and our typical refrigerators and heat pumps operate well above it.

The efficiency of thermodynamic machines is most often explained by starting with an ideal process using an ideal substance – using a perfect gas as a refrigerant that runs in a closed circuit. (For more details see pointers in the Further Reading section below). The gas would be expanded at a low temperature. This low temperature is constant as heat is transferred from the heat source to the gas. At a higher temperature the gas is compressed and releases heat. The heat released is the sum of the heat taken in at lower temperatures plus the electrical energy fed in to the compressor – so there is no violation of energy conservation. In order to ‘jump’ from the lower to the higher temperature, the gas is compressed – by a compressor run on electrical power – without exchanging heat with the environment. This process is repeating itself again and again, and with every cycle the same heat energy is released at the higher temperature.

In defining the coefficient of performance the energy from the heat source is omitted, in contrast to the electrical energy:

COP = \frac {\text{Heat released at higher temperature per cycle}}{\text{Electrical energy fed into the compressor per cycle}}

The efficiency of a heat pump is the inverse of the efficiency of an ideal engine – the same machine, running in reverse. The engine has an efficiency lower than 1 as expected. Just as the ambient energy fed into the heat pump is ‘free’, the related heat released by the engine to the environment is useless and thus not included in the engine’s ‘output’.

100 1870 (Voitsberg steam power plant)

One of Austria’s last coal power plants – Kraftwerk Voitsberg, retired in 2006 (Florian Probst, Wikimedia). Thermodynamically, this is like ‘a heat pump running in reverse. That’s why I don’t like when a heat pump is said to ‘work like a refrigerator, just in reverse’ (Hinting at: The useful heat provided by the heat pump is equivalent to the waste heat of the refrigerator). If you run the cycle backwards, a heat pump would become sort of a steam power plant.

The calculation (see below) results in a simple expression as the efficiency only depends on temperatures. Naming the higher temperature (heating water) T1 and the temperature of the heat source (‘environment’, our water tank for example) T….

COP = \frac {T_1}{T_1-T_2}

The important thing here is that temperatures have to be calculated in absolute values: 0°C is equal to 273,15 Kelvin, so for a typical heat pump and floor loops the nominator is about 307 K (35°C) whereas the denominator is the difference between both temperature levels – 35°C and 0°C, so 35 K. Thus the theoretical COP is as high as 8,8!

Two silly examples:

  • Would the heat pump operate close to absolute zero, say, trying to pump heat from 5 K to 40 K, the COP would only be
    40 / 35 = 1,14.
  • On the other hand, using the sun as a heat source (6000 K) the COP would be
    6035 / 35 = 172.

So, as heat pump owners we are lucky to live in an environment rather hot compared to absolute zero, on a planet where temperatures don’t vary that much in different places, compared to how far away we are from absolute zero.

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Further reading:

Richard Feynman has often used unusual approaches and new perspectives when explaining the basics in his legendary Physics Lectures. He introduces (potential) energy at the very beginning of the course drawing on Carnot’s argument, even before he defines force, acceleration, velocity etc. (!) In deriving the efficiency of an ideal thermodynamic engine many chapters later he pictured a funny machine made from rubber bands, but otherwise he follows the classical arguments:

Chapter 44 of Feynman’s Physics Lectures Vol 1, The Laws of Thermodynamics.

For an ideal gas heat energies and mechanical energies are calculated for the four steps of Carnot’s ideal process – based on the Ideal Gas Law. The result is the much more universal efficiency given above. There can’t be any better machine as combining an ideal engine with an ideal heat pump / refrigerator (the same type of machine running in reverse) would violate the second law of thermodynamics – stated as a principle: Heat cannot flow from a colder to a warmer body and be turned into mechanical energy, with the remaining system staying the same.

KarnoyiCikl

Pressure over Volume for Carnot’s process, when using the machine as an engine (running it counter-clockwise it describes a heat pump): AB: Expansion at constant high temperature, BC: Expansion without heat exchange (cooling), CD: Compression at constant low temperature, DA: Compression without heat exhange (gas heats up). (Image: Kara98, Wikimedia).

Feynman stated several times in his lectures that he does not want to teach history of physics or downplayed the importance of learning about history of science a bit (though it seems he was well versed in – as e.g. his efforts to follow Newton’s geometrical prove of Kepler’s Laws showed). For historical background of the evolution of Carnot’s ideas and his legacy see the the definitive resource on classical thermodynamics and its history – Peter Mander’s blog carnotcycle.wordpress.com:

What had puzzled me is once why we accidentally latched onto such a universal law, using just the Ideal Gas Law.The reason is that the Gas Law has the absolute temperature already included. Historically, it did take quite a while until pressure, volume and temperature had been combined in a single equation – see Peter Mander’s excellent article on the historical background of this equation.

Having explained Carnot’s Cycle and efficiency, every course in thermodynamics reveals a deeper explanation: The efficiency of an ideal engine could actually be used as a starting point defining the new scale of temperature.

Temperature scale according to Kelvin (William Thomson)

Carnot engines with different efficiencies due to different lower temperatures. If one of the temperatures is declared the reference temperature, the other can be determined by / defined by the efficiency of the ideal machine (Image: Olivier Cleynen, Wikimedia.)

However, according to the following paper, Carnot did not rigorously prove that his ideal cycle would be the optimum one. But it can be done, applying variational principles – optimizing the process for maximum work done or maximum efficiency:

Carnot Theory: Derivation and Extension, paper by Liqiu Wang

Mastering Geometry is a Lost Art

I am trying to learn Quantum Field Theory the hard way: Alone and from text books. But there is something harder than the abstract math of advanced quantum physics:

You can aim at comprehending ancient texts on physics.

If you are an accomplished physicist, chemist or engineer – try to understand Sadi Carnot’s reasoning that was later called the effective discovery of the Second Law of Thermodynamics.

At Carnotcycle’s excellent blog on classical thermodynamics you can delve into thinking about well-known modern concepts in a new – or better: in an old – way. I found this article on the dawn of entropy a difficult ready, even though we can recognize some familiar symbols and concepts such as circular processes, and despite or because of the fact I was at the time of reading this article a heavy consumer of engineering thermodynamics textbooks. You have to translate now unused notions such as heat received and the expansive power into their modern counterparts. It is like reading a text in a foreign language by deciphering every single word instead of having developed a feeling for a language.

Stephen Hawking once published an anthology of the original works of the scientific giants of the past millennium: Corpernicus, Galieo, Kepler, Newton and Einstein: On the Shoulders of Giants. So just in case you googled for Hawkins – don’t expect your typical Hawking pop-sci bestseller with lost of artistic illustrations. This book is humbling. I found the so-called geometrical proofs most difficult and unfamiliar to follow. Actually, it is my difficulties in (not) taming that Pesky Triangle that motivated me to reflect on geometrical proofs.

I am used to proofs stacked upon proofs until you get to the real thing. In analysis lectures you get used to starting by proving that 1+1=2 (literally) until you learn about derivatives and slopes. However, Newton and his processor giants talk geometry all the way! I have learned a different language. Einstein is most familiar in the way he tackles problems though his physics is on principle the most non-intuitive.

This amazon.com review is titled Now We Know why Geometry is Called the Queen of the Sciences and the reviewer perfectly nails it:

It is simply astounding how much mileage Copernicus, Galileo, Kepler, Newton, and Einstein got out of ordinary Euclidean geometry. In fact, it could be argued that Newton (along with Leibnitz) were forced to invent the calculus, otherwise they too presumably would have remained content to stick to Euclidean geometry.

Science writer Margaret Wertheim gives an account of a 20th century giant trying to recapture Isaac Newton’s original discovery of the law of gravitation in her book Physics on the Fringe (The main topic of the book are outsider physicists’ theories, I have blogged about the book at length here.).

This giant was Richard Feynman.

Today the gravitational force, gravitational potential and related acceleration objects in the gravitational fields are presented by means of calculus: The potential is equivalent to a rubber membrane model – the steeper the membrane, the higher the force. (However, this is not a geometrical proof – this is an illustration of underlying calculus.)

Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola

Model of the gravitational potential. An object trapped in these wells moves along similar trajectories as bodies in a gravitational field. Depending on initial conditions (initial position and velocity) you end up with elliptical, parabolic or hyperbolic orbits. (Wikimedia, Invent2HelpAll)

(Today) you start from the equation of motion for a object under the action of a force that weakens with the inverse square of the distance between two massive objects, and out pops Kepler’s law about elliptical orbits. It takes some pages of derivation, and you need to recognize conic sections in formulas – but nothing too difficult for an undergraduate student of science.

Newton actually had to invent calculus together with tinkering with the law of gravitation. In order to convince his peers he needed to use the geometrical language and the mental framework common back then. He uses all kinds of intricate theorems about triangles and intersecting lines (;-)) in order to say what we say today using the concise shortcuts of derivatives and differentials.

Wertheim states:

Feynman wasn’t doing this to advance the state of physics. He was doing it to experience the pleasure of building a law of the universe from scratch.

Feynman said to his students:

“For your entertainment and interest I want you to ride in a buggy for its elegance instead of a fancy automobile.”

But he underestimated the daunting nature of this task:

In the preparatory notes Feynman made for his lecture, he wrote: “Simple things have simple demonstrations.” Then, tellingly, he crossed out the second “simple” and replaced it with “elementary.” For it turns out there is nothing simple about Newton’s proof. Although it uses only rudimentary mathematical tools, it is a masterpiece of intricacy. So arcane is Newton’s proof that Feynman could not understand it.

Given the headache that even Corpernicus’ original proofs in the Shoulders of Giants gave me I can attest to:

… in the age of calculus, physicists no longer learn much Euclidean geometry, which, like stonemasonry, has become something of a dying art.

Richard Feynman has finally made up his own version of a geometrical proof to fully master Newton’s ideas, and Feynman’s version covered hundred typewritten pages, according to Wertheim.

Everybody who indulges gleefully in wooden technical prose and takes pride in plowing through mathematical ideas can relate to this:

For a man who would soon be granted the highest honor in science, it was a DIY triumph whose only value was the pride and joy that derive from being able to say, “I did it!”

Richard Feynman gave a lecture on the motion of the planets in 1964, that has later been called his Lost Lecture. In this lecture he presented his version of the geometrical proof which was simpler than Newton’s.

The proof presented in the lecture have been turned in a series of videos by Youtube user Gary Rubinstein. Feynman’s original lecture was 40 minutes long and confusing, according to Rubinstein – who turned it into 8 chunks of videos, 10 minutes each.

The rest of the post is concerned with what I believe that social media experts call curating. I am just trying to give an overview of the episodes of this video lecture. So my summaries do most likely not make a lot of sense if you don’t watch the videos. But even if you don’t watch the videos you might get an impression of what a geometrical proof actually is.

In Part I (embedded also below) Kepler’s laws are briefly introduced. The characteristic properties of an ellipse are shown – in the way used by gardeners to creating an elliptical with a cord and a pencil. An ellipse can also be created within a circle by starting from a random point, connecting it to the circumference and creating the perpendicular bisector:

Part II starts with emphasizing that the bisector is actually a tangent to the ellipse (this will become an important ingredient in the proof later). Then Rubinstein switches to physics and shows how a planet effectively ‘falls into the sun’ according to Newton, that is a deviation due to gravity is superimposed to its otherwise straight-lined motion.

Part III shows in detail why the triangles swept out by the radius vector need to stay the same. The way Newton defined the size of the force in terms of parallelogram attached to the otherwise undisturbed path (no inverse square law yet mentioned!) gives rise to constant areas of the triangles – no matter what the size of the force is!

In Part IV the inverse square law in introduced – the changing force is associated with one side of the parallelogram denoting the deviation from motion without force. Feynman has now introduced the velocity as distance over time which is equal to size of the tangential line segments over the areas of the triangles. He created a separate ‘velocity polygon’ of segments denoting velocities. Both polygons – for distances and for velocities – look elliptical at first glance, though the velocity polygon seems more circular (We will learn later that it has to be a circle).

In Part V Rubinstein expounds that the geometrical equivalent of the change in velocity being proportional to 1 over radius squared times time elapsed with time elapsed being equivalent to the size of the triangles (I silently translate back to dv = dt times acceleration). Now Feynman said that he was confused by Newton’s proof of the resulting polygon being an ellipse – and he proposed a different proof:
Newton started from what Rubinstein calls the sun ‘pulsing’ at the same intervals, that is: replacing the smooth path by a polygon, resulting in triangles of equal size swept out by the radius vector but in a changing velocity.  Feynman divided the spatial trajectory into parts to which triangles of varying area e are attached. These triangles are made up of radius vectors all at the same angles to each other. On trying to relate these triangles to each other by scaling them he needs to consider that the area of a triangle scales with the square of its height. This also holds for non-similar triangles having one angle in common.

Part VI: Since ‘Feynman’s triangles’ have one angle in common, their respective areas scale with the squares of the heights of their equivalent isosceles triangles, thus basically the distance of the planet to the sun. The force is proportional to one over distance squared, and time is proportional to distance squared (as per the scaling law for these triangles). Thus the change in velocity – being the product of both – is constant! This is what Rubinstein calls Feynman’s big insight. But not only are the changes in velocity constant, but also the angles between adjacent line segments denoting those changes. Thus the changes in velocities make up for a regular polygon (which seems to turn into a circle in the limiting case).

Part VII: The point used to build up the velocity polygon by attaching the velocity line segments to it is not the center of the polygon. If you draw connections from the center to the endpoints the angle corresponds to the angle the planet has travelled in space. The animations of the continuous motion of the planet in space – travelling along its elliptical orbit is put side-by-side with the corresponding velocity diagram. Then Feynman relates the two diagrams, actually merges them, in order to track down the position of the planet using the clues given by the velocity diagram.

In Part VIII (embedded also below) Rubinstein finally shows why the planet traverses an elliptical orbit. The way the position of the planet has finally found in Part VII is equivalent to the insights into the properties of an ellipse found at the beginning of this tutorial. The planet needs be on the ‘ray’, the direction determined by the velocity diagram. But it also needs to be on the perpendicular bisector of the velocity segment – as force cause a change in velocity perpendicular to the previous velocity segment and the velocity needs to correspond to a tangent to the path.

Surprise Potatoes in the Soldiers’ Vegetable Soup!

Having blogged for more than a year I have finally reached the status of renowned, serious blogger. I have carved out my niche, and I have been asked for providing feedback on a book in that particular category.

Of course, it is a book of spam poems.

 Surprise Potatoes in the Soldiers’ Vegetable Soup

… compiled by my LinkedIn connection Alan Mundy – “poorly translated Chinese recipes cannibalised to form the most insightful and thought provoking book of its kind ever written (presumably)”

Checking Alan’s LinkedIn URL again I confirm it starts with uk which does not come as a surprise. A book like this can only originate from the country that produced Shakespeare, Monty Python and Douglas Adams.

I am a blog spam expert, so it is an enormous task to review, understand, and do justice to e-mail spam poetry.

Meta-Information

A bunch of spam poets – Alan Mundy, Jess Bryan, Rob Cleaver, Richard Sutton, and Dan Roberts (If any of you wants to have your name sanitized for the sake of online reputation – let me know and I replace it with *****; if I have forgotten somebody let me know, too) – have assembled poems from spam, adhering to the following rules:

  • You can only use lines from the text in your poem – you must not add anything
  • You must not edit the original lines in any way
  • You can use partial lines but must not mix lines together to create new lines

(I promise I will follow these next time, too!)

The book contains 101 spam poems plus the original spam e-mails as bonus material, sort of ‘making of’. The original e-mail spams are rather long-winded which might give the poet a greater selection of phrases to pick from, but in the other hand it might be tiresome to read through all this without turning your brain into the juice of three bark.

Poetry

The poems are as food-centric as the original spam was. This was a novel experience for the philosophically inclined geek in me who prefers postmodern spam poems lingering on the new age-y.

Having read the book for countless times in the past week I have changed my mind – though recommended to all the hobby chefs among you it the poems will also appeal to the refined ethereal poetry lovers. The poems contain gems of timeless wisdom such as Very often you use also be young and aphorisms on ethics such as Be good if you die.

It is in particular the embroidery with all stuff food-related that provides a consistent down-to-earth theme to put all these grand insights gained from spam into perspective. So the poems are both artistic as comprehensible to readers that did not have that much exposure to advanced experimental poetry. After all didn’t great physicist Richard Feynman say A poet once said, ‘The whole universe is in a glass of wine.’? Cross-checked again: Wine is featured in four poems!

Also the Stephen King fans will enjoy their share of creepy violence – Cut in your liver!  – and science geeks will love terms as Transmission intensity.

There is also a poem titled A sexual poem… (The header lines have been created by the poets BTW). Thanks for your understanding that I cannot quote from this on your geeky family blog though it might boost my Google ranking.

So I give this alleged first(*) book on spam poetry 5 of 5 stars.
(*) As usual, I did not do research on this, and I do not want to be involved in disputes about originality. It’s probably the spammers who own the stuff and who have licenced it under Creative Commons.

Vaina chilena - Mario Gonzalez

Since there is a lot of yolk in these poems and some wine, and since a glass of wine contains the universe according to Richard Feynman, I have picked an image of a cocktail containing both (Trusting Wikipedia / Wikimedia on this).

PS: I have not forgotten about  my scheduled post on networking, professional online profiles and the like. But now you know already how professionals really use LinkedIn!
PPS: Calling people ‘connections’ is LinkedIn’s terminology, not mine.
PPPS: My blog spam queue is exploding!


Is It Determinism if We Can Calculate Probabilities Exactly?

I set a stretch goal for myself: I want to force myself to keep some posts of mine short.

As a fan of MinutePhysics I am launching a new category: Physics in a Nutshell. I am going to try to tackle a question that has bothered me for a while – hopefully briefly and concisely.

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The question of today is:

Butterfly Effect

The Infamous Butterfly. Part of Pop Culture, at least since we had all watched Jurassic Park (Wikimedia)

What is determinism (in physics)?

The extremely short answer is: I don’t know. Or rather: I don’t know if the question makes so much sense as first of all we need to get terminology right and work out a 1 on 1 map between mathematical and philosophical terms. This is an issue I have with those philosophical questions in physics – I am rather a fan of Shut up and Calculate and of Richard Feynman’s pragmatic attitude in general (whether or not the quote can be attributed to him).

Usually we say a system is deterministic if we can predict its future (calculate any property we are interested in) based on its current properties and some fundamental laws.

Classical mechanical systems are deterministic on the fundamental level in the sense that we could predict their behaviors on principle. On principle only, because prediction means calculations, and calculations might be subject to tiny errors adding up when systems turn chaotic. Think turbulence, weather and the infamous butterfly.

Quantum systems are non-deterministic in the sense that the fundamental laws in quantum theory do not give us what we (classically minded) consider the ‘full information’. Think momentum or position versus both, or think entangled particles – we only know both would show the same quantum number when measured, but we do not know which number.

But depending on the system you consider (forces, number of ‘particles’) the quantum mechanical equations may not exhibit chaotic behaviour. So in this case you calculate probabilities, but these probabilities you will know for sure.

We do not even need to allude to spooky quantum stuff – just consider a classical system comprising tons of particles, such as a the air in a room. In this case of a messy classical system we can only forecast probabilities for practical reasons, but statistically defined properties are all that matters anyway. Think temperature.

In real, interesting systems involving lots of forces, matter and stuff, you will encounter both aspects of non-determinism. If you want to tag it that way. I would rather stick with equations, boundary conditions, stability of numerical calculations and other technical terms.

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Recommended reading

Richard Feynman, Vol. I of his Physics Lectures. Though sub-titled Mainly Mechanics, Radiation, and Heat it is really about all of physics, including quantum mechanics.

Loosely connected remarks

Ironically, despite Feynman’s explicit depreciation of useless armchair philosophy and reluctance of well-rounded education in liberal arts, many of his statements comprise profound philosophical truths. Sometimes tucked away in a footnote:

Poets say science takes away from the beauty of the stars – mere globs of gas atoms.  Nothing is “mere.”  I too can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination – stuck on this carousel my little eye can catch one million year old light… What is the pattern, or the meaning, or the why?  It does not do harm to the mystery to know a little about it.  For far more marvelous is the truth than any artists of the past imagined!  Why do the poets of the present not speak of it?  What men are poets who can speak of Jupiter as if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?”

(R. Feynman, physics Lectures, Vol.I, Footnote in section 3-4)

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This post was inspired by many current discussions, finally triggered most likely by this discussion on LinkedIn

The Spinning Gyroscope and Intuition in Physics

Antique spinning topIf we would set this spinning top into motion, it would not fall, even if its axis would not be oriented perpendicular to the floor. Instead, its axis would change its orientation slowly. The spinning motion seems to stabilize the gyroscope, just as the moving bicycle is sort of stabilized by its turning wheels. This sounds simple and familiar, but can this really be grasped by intuition immediately?

I do not think so – otherwise it would not have taken us 2000 years to get over Aristotle’s assumptions on motion and rest. And simple experiments demonstrated in science shows would not baffle us – such as the motion of a helium balloon in an accelerating car.

The standard text-book explanation goes like this: There is gravity, as we assume that the spinning top is not supported in its center of gravity. Thus there is a torque. The gyroscope is whirling, thus it has angular momentum. A torque corresponds to a change in angular momentum, analogous to a force resulting in a change of (linear) momentum. The torque vector is perpendicular to gravity and to the axis of the gyroscope. Thus the change in angular momentum is always perpendicular to the current angular momentum vector and the tip of the spinning top moves in a circle. The angular momentum vectors changes all the time – not in length, but in direction – which is called precession.

As Richard Feynman pointed out in his Physics Lectures, this explanation constitutes rather mathematical step-by-step instructions than a real explanation. We do not see immediately why the spinning top precesses instead of falling to the ground.

Our skepticism is justified: The text-book explanation does not fully expound the dynamics of the systems and explain what really happens – in the very moment the spinning top starts to move. It rather refers to a self-consistent solution: If the gyroscope would already precess in a circle, that circular movement is consistent with the torque. As everybody in his right mind (R. Feynman) would assume, it actually might fall a bit if it is released.

Spinning Top - PrecessionGenerally, the tip of the gyroscope keeps tracing out a wavy or loopy path, which is called nutation.

If the spinning top nutates / starts falling, it looses potential energy. This has to compensated by an increase in rotational energy, the velocity of the tip of the gyroscope is not a constant. (Note that the total angular momentum of the gyroscope is composed of contributions from the fast spinning motion and the slow precession). The tip of the gyroscope moves on a curved trajectory bending upwards, which finally leads to overshooting the average height.

Friction can make the wobbling decay and finally turn the trajectory into the simple-text-book-path. This simulation allows for turning on friction (which is also equivalent to Feynman’s explanation).

An excellent explanation can be found in this remarkable paper (related to the simulation): The gyroscope is set into rotational motion while still supported. When “gravity is suddenly turned on” by removing the support, the additional vertical component of the angular momentum – due to to precession – is suddenly turned on. The point is that the initial angular momentum is parallel to the symmetry axis of the gyroscope, and the axis starts from velocity zero.  The total angular momentum – still parallel to the symmetry axis – is the sum of the one related to precession and the one related to the gyroscope’s fast movement. So the latter is not parallel to the axis any more: The tip of the axis starts tracing out the loopy path (nutation) when it precesses. Only if we tune the angular frequency carefully before we release the spinning top, the text-book solution can be obtained. In this case precession is really maintained by the torque.

So do we understand the gyroscope intuitively now? A deep understanding of angular momentum and torque is a pre-requisite in my point of view. On principle, all of classical mechanics can be derived from Newton’s laws, so the notions of force and momentum should be sufficient. Nevertheless, without introducing angular momentum, there is no way to explain the motion of the gyroscope briefly.

Why do we need “torque” in general? Such concepts are shortcuts that allow for a concise description, but they also reveal the underlying symmetry or essential aspects of a problem. You could describe the dynamics of a rigid body by considering the motion of all little pieces the body is composed of. But since it is rigid, actually two points would be sufficient. You can select any two points or basically any set of independent coordinates – 6 independent numbers.

The preferred choice is: 3 numbers – such as Cartesian co-ordinates, x,y,z – describing the motion of the center of gravity and 3 numbers describing the rotation of the body. You need two numbers to denote the direction of the axis about which to rotate (similar to two longitude and latitude to describe a point on a sphere), and one number to denote the angle – how much you rotate. You could also describe any rotation in terms of the components discussed for the gyroscope: precession, nutation and internal rotation.

Then Newton’s equation of motion for the rigid body can be re-written as a law of motion for the center of gravity (Force equal change of momentum of the center of gravity) and a law for two new properties of the system: the torque equals the change of the angular momentum. Actually, this equation defines what these properties really are. Checking the definitions that have evolved from the law of motion we conclude that the angular momentum is linear momentum times the lever arm and the torque is force times lever arm. But these definitions as such would not make sense if they would not have been generated by the reformulation of Newton’s law.

I think we sometimes adopt or memorize definitions carelessly and consider this learning because these definitions are required by standards / semi-legal requirements and used within a specific community of experts. But there is no shortcut and no replacement of understanding by learning definitions by heart.

I believe you need to keep the whole entangled web of relations between fundamental laws and absolutely necessary quantities in mind, but it is hard how to restrict the scope. We could now advance from gyroscope and angular momentum to the deeper connections between symmetries and conservation laws. In order not get stuck in these philosophical musings all the time – and do something useful (e.g. as an engineer), you need to be able to switch to shut-up-and-calculate-mode (‘Shut up and calculate’ is often attributed to Richard Feynman, but I could not find an authoritative confirmation).

Sniffing the Path (On the Fascination of Classical Mechanics)

Newton’s law has been superseded by relativity and quantum mechanics, and our universe is strange and compelling from a philosophical perspective. Classical Mechanics is dull.

I do not believe that.

The fundamentals of Newtonian Mechanics can be represented in a way that is different from well-known Force = Mass Times Acceleration – being mathematically equivalent, but providing a different philosophical twist. I consider this as fascinating as the so-called spooky action-at-a-distance of quantum mechanics.

The standard explanation is this:

  • There are forces described by respective laws (e.g.: The gravitational force)
  • Forces act on matter and result in the acceleration of particles.
  • In every point of time, the path of the particle can  be calculated based on its acceleration if you know its location and its velocity in the point of time before.
  • Thus step by step, the particle explores its path and the final trajectory is composed of all these tiny steps.

This is why the classical world seems to be deterministic. (Yes, this explanation lacks the interdependence of space(time) and masses and the limitations imposed by quantum mechanics.)

The deterministic laws can be stated in terms of The Principle of Least Action

  • Consider the point in space where the particle starts off and the end point of the journey. Thus we look at the path in hindsight: We demand that the particle needs to travel from A to B, and we also fix the points of time.
  • Now we evaluate all possible trajectories the particle might travel from A to B.
  • For every path we calculate a number: This is called the “Action” (In simple particle mechanics this is equivalent to integration over the difference of kinetic and potential energy – the “action” isn’t something you can easily “feel” like the force or the momentum). Note that the total energy needs to be conserved, thus e.g. there should be no friction. But at a microscopic level, all forces are conservative anyway.
  • Above all: Note that a single number is assigned to a full path which consists of all the points in space the particle traverses.
  • The path that is actually traversed / realized is the path that us assigned the least action.

Thus is seems that the particle sniffs all the paths ((c) Richard Feynman) and selects a path distinguished by a particular property. In addition, we have replaced the necessity to know the initial location and velocity by the knowledge of the location at the start point and the end point.

It seems we are (nature is) working backwards.

Actually, the particle is really sort of sniffing the path: This is a minimum, exactly: an extremum. Near an extremum the slope of a function is nearly zero. Thus the particle sniffs the neighbouring paths and checks for changes in the action. The apparent contradiction between working forward and backwards is resolved if The Principle of Least Action is applied to smaller and smaller pieces of the trajectory. Since the principle holds for any path, it also needs to true for infinitesimal parts of a path. For these infinitesimal paths, the principle boils down (mathematically) to Newton’s law.

A mathematical derivation might not be satisfactory from a philosophical point of view. Probably the following may serve as an explanation: Working with the Principle of Least Action we do not know or do not need to know the velocity at the start time. Thus we need some other information instead. By the way, also data as the total energy, momentum or angular momentum may be used as a substitute of the total information about the initial conditions in terms of position and velocity.

Using the Principle, we know only where we are heading for. Since we do not know the initial velocity – the tangent to our path at start time – we need more guidance. The Principle provides such guidance and allows the particle for sniffing for other paths in order to determine that tangent at every point of time.

Unification of Two Phenomena Well Known

Unification is a key word that invokes some associations: The Grand Unified Theory and Einstein’s unsuccessful quest for it, of course the detection of the Higgs boson and the confirmation of the validity of the Standard Model of Particle Physics, or Kepler’s Harmonices Mundi.

Unification might be driven by the search for elegance and simplicity in the universe. Nevertheless, in retrospect it might be presented as down-to-earth and straight-forward.

Electricty and magnetism have been considered distinct phenomena until they have been “unified” by describing them by Maxwell’s equation that are consistent with the theory of relativity. What does this mean?

I am summarizing the explanation given by Richard Feynman in chapter 13 of volume II of his Physics Lectures:

Consider a wire carrying an electric current and a small test charge near the wire. The test charge moves with constant velocity and follows a path parallel to the wire. The wire comprises positive ions and free electrons and is this electrically neutral, so it exerts no electrical force on the test charge. However, the motion of the electrons gives rise to a magnetic field. Since the test charge is moving, the magnetic field gives rise to a force (the Lorentz force) that makes the charge move in a direction perpendicular to the wire (the sign depends on the type of charge. A negative charge would be attracted to the wire if it travels in the same direction as the electrons in the wire).

Now imagine you would watch this experiment from the perspective of an observer who moves with a velocity equal to the velocity of the test charge. The charge is now at rest. If the test charge had moved with exactly the same speed as the electrons before (this is an assumptions made for the sake of simplicity), from the travelling observer’s perspective the electrons in the wire would be at rest and the positive ions would be moving. So since some carriers of charge do still move, a magnetic field would also exist in that frame of reference. However, the field would not exert a magnetic force on the test charge that is now standing still.

If the charge would move towards the wire an eventually hit it in one frame of reference, the same effect needs to be observed in the other. What kind of force would be accountable for that in the second frame of reference?

It is an electrical force and it is due to the fact that the wire is electrically charged in the second system. Electrical charges of particles do not change with switching to different inertial frame, but dimensions parallel to the relative velocity do. And thus does the charge density – the charge per unit volume or per unit length of the wire. If there a charge density ρ  is measured by an observer at rest, the observer in motion relative the charges measures a larger charge density because the volume has shrunk by a factor of √(1 – v²/c²) (This is the infamous factor appearing in all kinds of equations in relativity, c being the speed of light in vacuum). If charge density changes, there is a net overall charge per unit volume.

Why do the swap of the roles of positive and negative charges not compensate for that? The travelling electrons turned to static charges and the static ions turned to moving positive charges. Remember that the wire is electrically neutral in the system considered first. Thus in this system the charge density of electrons is larger than their density measured in the travelling system. Switching to the latter system, the correction factors are applied to each type of charge in a different way – starting from the densities measured in the system “at rest”: The ion density is increased as these are moving now, but the electron charge density is reduced, as we have measured the increased density in the other system.

Actually, the forces turn out be different by a factor equal to the square root mentioned above, the force is smaller for system 2. But this is needed for consistency: The effect of the force is measured by its impact – its momentum. In special relativity the momentum is often illustrated by the penetration depth of a bullet (driven into some material, in a direction perpendicular to the relative velocity). Momentum is force times the interval of time the force is acting on a particle. But time is dilated according to special relativity, that is: time intervals appear longer if the particle is moving (system 1). Thus by calculating the product of force and time interval, the factors cancel out exactly.

In summary, the forces of electricity and magnetism morph into each other – dependent on the frame of reference chosen. They are two aspects of some underlying “unified” force. On the one hand, this changed the way we think about the electromagnetism.

On the other hand – technically it just means that we take the components of electrical and magnetic fields (3 numbers each – these are vectors) and stuff them into some more general mathematical structure consisting of 6 numbers (This is called a tensor). This sounds simple and there is a reason for that: Historically, Maxwell’s equation that govern the spatial and temporal evolvement of electrical and magnetic fields have been laid down before Einstein developed the theory of special relativity. Maxwell’s equations had already been consistent with special relativity and they did not need amendment – as Newton’s law. So unification did exist already – mathematically, but the consequences had not been fully understood.