What's the minimum (physics first) to get an oscillator?

[RoGeorge]

So, it's the particular arrangement of a given set of rules, plus the interaction between them, and that makes a certain behavior to emerge.

In the physical world, the set of rules would be the laws of physics.  And to make a particular arrangement out of them, it usually means to put some in close proximity, such that those laws will be interacting with each other.  There's so much to speculate in this direction.

The idea of everything taken as emergent behavior feels like an epiphany, because until now I was considering as emergent phenomena only the appearance of something that it is not really there, for example a bank of fish being "repelled" by a predator.

[TimFox]

In physics, the harmonic oscillator in its various forms (simple, damped, forced, coupled, etc.) is fundamental to many other phenomena, and freshman physics concentrates on it accordingly.
Often, analyzing such systems in terms of energy is easier to understand than an analysis in terms of forces, but the physical results are the same.
In advanced classical physics,  the Hamiltonian re-formulation of Newtonian mechanics is a very powerful tool to deal with more complex systems with many degrees of freedom, defining a "generalized co-ordinate" and "generalized momentum" for each degree of freedom (which may not be perpendicular to each other), and later work used the Hamiltonian approach for quantum mechanics.
A good senior-undergraduate textbook for such topics is by Goldstein  https://physicsgg.files.wordpress.com/2014/12/classical_mechanics_goldstein_3ed.pdf
See Chapter 6 for a detailed description of oscillation.
(When I purchased the first edition back in 1969, I believe it cost $17.  Much later, I purchased the third edition for approximately $110.  The above link is a downloadable huge pdf.)

[CatalinaWOW]

So, it's the particular arrangement of a given set of rules, plus the interaction between them, and that makes a certain behavior to emerge.

In the physical world, the set of rules would be the laws of physics.  And to make a particular arrangement out of them, it usually means to put some in close proximity, such that those laws will interacting with each other.  There's so much to speculate in this direction.

The idea of everything taken as emergent behavior feels like an epiphany, because until now I was considering as emergent phenomena only the appearance of something that it is not really there, for example a bank of fish being "repelled" by a predator.

You are in some deep philosophical ground there.  Newton's laws, the spring constant relationship and other rules are our description of how the system works.  We like the description because it matches behavior.  But I don't think the rules make the system.  The system is, and happens to conform to some degree to a set of rules we have constructed.  We may or may not understand the underlying system.

The most common example to make this point clearer is the use of epicycles to describe the motions of heavenly bodies in a geocentric model of the cosmos.  These rules actually worked quite well, and described the behavior of the visible bodies in the sky to the limits of observational accuracy for millenia.  It wasn't until about 500 years ago when the very detail measurements of Tycho Brahe started to show discrepancies with the simpler epicycle models.  And more complex epicycle models were constructed to explain the new data.   Today most would agree that actual reality doesn't conform to these sets of rules.

The glib answer to your question of why the energy transfers back and forth is because nothing is stopping it.  If you connect a charged capacitor to an inductor what is stopping that charge (and hence the stored energy) from flowing through the inductor?   In this case the inductor puts up a bit of a fight, generating a counter voltage - the back emf - proportional to the change in current.  But it only slows, does not stop the flow.  At some point the voltage on the capacitor drops below the back emf and the sign of the current change reverses.  Now the inductor is pulling charge out of the capacitor to prevent instantaneous magnetic field collapse.  Which continues until the field reaches zero, where it turns out that all of the charge has returned to the capacitor.

Same exact situation as releasing a mass attached to a stretched spring.  Nothing is stopping the mass from moving so the spring has its way.  For a while.

[jwet]

Quartz crystals also have this bouncing feature- strain to electricity, electricity to strain.  Its not a necessity of the physics but it does make for an "efficient" oscillator.

[TimFox]

An interesting difference between resonators such as quartz crystals and simple oscillators (mass and spring) is that resonators can have multiple resonant frequencies, while simple oscillators have only the one.
One-dimensional resonators (organ pipes, violin strings, resonant transmission lines, etc.) have harmonic overtone frequencies (integer multiples of the lowest or fundamental resonant frequency), but three-dimensional resonators (quartz crystals, resonant cavities, etc.)  have overtones that are not harmonic frequencies.

[AndyBeez]

The prototype harmonic oscillator in physics discussions is a spring

How about a pendulum ?  Or does the externality of gravity complicate things?

Or a planet orbiting a star? Velocity relative to the star comes out as a sine wave... of sorts. Also electrons orbiting the nucleus too. For a pure audio sine wave, how about the physics of a tuning fork? Really, everything is in oscillation. Or resonating. Except for that RF oscillator you designed and spent hours building

[T3sl4co1l]

An interesting difference between resonators such as quartz crystals and simple oscillators (mass and spring) is that resonators can have multiple resonant frequencies, while simple oscillators have only the one.
One-dimensional resonators (organ pipes, violin strings, resonant transmission lines, etc.) have harmonic overtone frequencies (integer multiples of the lowest or fundamental resonant frequency), but three-dimensional resonators (quartz crystals, resonant cavities, etc.)  have overtones that are not harmonic frequencies.

Sort of.  Several of these are dispersive so have nonharmonic overtones.  Acoustic resonators are a good example, or better to say: acoustic waveguides.  Most waveguides I suppose, share this property?

Bulk material vibrations are especially rich, having three directions each longitudinal, transverse, and shear, all with different wave velocities, all which couple to each other to varying degrees depending on geometry (and in nonlinear combinations, because displacement determines the moment on which longitudinal force acts, etc.).

EM structures with interaction between nearby elements (not just directly adjacent) also works.  Consider the helical waveguide for example.

Or for more quantum examples, consider the 1/n^2 spectrum of atomic levels; the resonator is highly irregularly shaped (a spherical 1/r potential well) so different states have very different waves.  At least until you get to the Rydberg sort of levels where states (or modest superpositions thereof) start to look like classical orbits.

Tim

[TimFox]

Also, in the classical simple harmonic oscillator, obviously the minimum possible energy stored in the system is zero, where the displacement and velocity are both zero.
In quantum mechanics, such a state violates Heisenberg, since both the position and momentum are zero, with no uncertainty.
In the quantum harmonic oscillator (a prototype for many physical systems), the minimum state is given by
E₀ = (h f)/2  , where f is the natural frequency of the oscillator and h is Planck's constant.
and the discrete set of energies is given by
E_n = (n + 1/2) h f
for non-negative integers n
The spacing between these states is h f
(This is usually written h-bar omega, but I'm too lazy for proper typesetting.)

[TimFox]

Separate issue about overtones and harmonics:
Sometimes people confuse the harmonic frequencies of a simple (non-dispersive) one-dimensional system (such as a violin string) with the harmonic frequencies that appear in the Fourier series for the decomposition of any periodic waveform.
They have the same form (integer times fundamental frequency), but originate differently.

[AndyBeez]

...and ring modulation

[jwet]

Further down the rabbit hole...

I worked on what was to be "the lowest power" Real Time Clock project at Maxim a while back.  Maxim was big on superlatives and did a lot of projects like this- we called them "10x" projects where you blew some important spec out of the water by 10x or more.  I was the product definer- an apps engineer that works with design to understand the need fully, nail down specs, often interview customers and do market and technical research.  One of things that came out of this for me in a philosophical sense was that its "impossible" to keep time without using energy.  Anything that keeps time uses energy, I thought this was interesting if not profound.  We delivered a 200 nA RTC- DS1302.  The world record today is held by the MAX31331- taking only 65 nA to keep time!  There are real needs for sub sub 1 uA clocks believe it or not.

[thermistor-guy]

...
One-dimensional resonators (organ pipes, violin strings, resonant transmission lines, etc.) have harmonic overtone frequencies (integer multiples of the lowest or fundamental resonant frequency), but three-dimensional resonators (quartz crystals, resonant cavities, etc.)  have overtones that are not harmonic frequencies.

Speaking of three-dimensional resonators: the earth-ionosphere cavity, excited by lightning:

https://en.wikipedia.org/wiki/Schumann_resonances

[metebalci]

A physics-first approach is asked, but I guess I am not a physics person, and I think about oscillations often in terms of dynamical systems recently. So I guess one needs a proper configuration of one or two fixed points, or a limit cycle or a strange attractor for oscillatory behavior. The behavior of latter two may quickly become weird so not many practical electronics applications I guess, other than sound/noise generation maybe.

[Nominal Animal]

So, it's the particular arrangement of a given set of rules, plus the interaction between them, and that makes a certain behavior to emerge.

Yes.  Cellular automata, especially Conway's Game of Life, is a good way of experimenting with this.  If you like combining low-level programming (large world, bit per cell) and user interfaces (viewing the world at 1:1 or more than one cell per pixel), even more so.  You can adjust the rules, including extend the grid area involved, add directional weights, et cetera.  Even with the Conway's trivial ruleset, it is Turing-complete: there are enough emergent behaviour that any Turing machine (computer) can be simulated.

Simulating even simple microprocessors takes huge numbers of cells, and therefore clever optimizations to compute the cell states efficiently; yet, it is always only the simple rules being implemented.  (It is also an interesting exercise in examining the rules themselves: sometimes some simple rule is very difficult to implement, whereas some complex rule can be trivial to implement.  We cannot really tell, until we attempt to implement the rule, somehow, first.)

The idea of everything taken as emergent behavior feels like an epiphany, because until now I was considering as emergent phenomena only the appearance of something that it is not really there, for example a bank of fish being "repelled" by a predator.

This is exactly how "vague" language can constrict thought, and why mathematics is so useful in describing the rules and behaviour.

Some insist that mathematics is not a language.  Perhaps so, but it definitely can be used as one to describe these things.

The true "trick" is correctly and efficiently translating between math and human understanding, especially human intuition.

[RoGeorge]

Spoken language may be too vague, while mathematics can be too exact, either of these extremes can be less productive.

Math is a great tool for crunching relations/abstractions, similar to how computers are good at crunching numbers.  Though, it is too exact to let room for any serendipity.  Serendipity helps a lot when exploring. 

I don't see mathematics as a seeking tool.  Only using it occasionally, as a confirmation tool to check if an idea is correct, or as a dismissing tool, to get rid of those too seductive hypotheses that stick to mind.

Then, I don't know enough mathematics.  It's hard to learn, hard to handle, mistakes are easy to make but hard to spot, it is crucial to properly encode, then decode/interpret the results, etc.

Most of all, it's an axiomatic system, that started from the real world.  Remember it all started from counting the sheep and measuring the land.  That also means it will not work outside its axioms.  There are always tacit assumptions, even in math axioms.  Having in mind the way geometry started from measuring flat agriculture land, and that led to Euclidean geometry.  If axioms are even slightly modified, we get new theorems and new mathematics, like the non Euclidean geometry.  Not a big deal now, looking in perspective, but a big deal if back then one would try to use Euclidean geometry to prove/disprove some non Euclidean hypothesis.  One would have get a big no, because of using the wrong set of axioms.

I guess these are all epistemology 101 and studied systematic there, though I never managed to follow any book or lectures about epistemology, looked very outdated.  I see human mind as a fuzzy lookup table, where the lookup table entries were filled up in the very early years, with notions we learn by interacting with the world.  Those are the basic concepts in which we understand everything else.  Later we learn to associate the most common entries in our lookup tables with words.  Filling the table happens in the very first years of life, before self-awareness emerges.  The table will keep updating for the rest of the life, but at a much slower peace.  When a fact matches very well the lookup table entries, we are in the ordinary, expected, or maybe boring situations.  When we encounter something for which we don't have a matching entry yet, we tend to reject that, or to take it as a "mind-blowing" new something.

All we do, and feel, and think is in terms of our inner lookup table, plus some internal noise.  The inner lookup tables being fuzzy and noisy, can sometimes lead to unexpected behavior or to creativity.

Sounds a lot like tokens from ChatGPT, but the fuzzy lookup tables idea came to me from hardware, not software.  I've start playing with the lookup table analogy a couple of decades ago, probably inspired from the FPGA architecture, where everything is a lookup table.

Even funnier when it comes to awareness.  I think what we call self-awareness is the ability of computing assisted-simulations of the world in which we live in, based on two things:
- the already existing entries in the fuzzy lookup tables
- the live sensory input data stream we have from our senses

By assisted-simulation, I mean the projections of the outcome is based on the lookup table, and the sensory input is used as a feedback.  Sensory input (e.g. vision, hearing, tactile, etc) is what we compare against, in order to check how accurate the prediction/simulation was.  Then adjust the simulation accordingly.  If we are left out without any sensory input, we start hallucinating (e.g. when sleeping, or under certain drugs, or in isolation tanks).

In fact, the more advanced an animal is in the tree of life, the more aware and the more capable is at computing the surroundings.  Reaction times are very slow, so animals got themselves good at predicting the future.  I believe our day to day life is a living simulation, a dream that is continuously adjusted, adjusted such that the simulation won't diverge too far from the sensory input stream.

We navigate reality, and react to what our mind predicted, rather than to what our senses read back from the environment.


Sorry for going with the flow with the offtopic, it's hard to be concise and convincing at the same time.  All these might be as well nothing but too much noise in the lookup tables.

Not trying to dismiss math, mathematics is a must have for rigorous work.  Though, when seeking/exploring at leisure, I found handling concepts from that fuzzy lookup table more fun to work with than math.  It's way faster to process inner thoughts, and more productive at generating new ideas.  Then triage later which ones may worth investigating further.

Many such generated ideas will be way off, and the internal lookup table keeps updating according to what it is exposed-to the most (that's a pitfall of our inner working, learning is always on, so a lie repeated a thousand times becomes truth).  So it's a risk that, eventually, some of the false ideas generated by words instead of math will be assimilated as truth in the own lookup table.  And just like that, one might start wearing pointy hats and pretend to be Napoleon. 

That's why one should never skip the mathematical proof.

[HuronKing]

On the subject of mathematics as descriptor though - I am heavily persuaded by the fact that mathematical constructs have led us to predicting and discovering pretty bizarre phenomena in nature - like the positron.

Like, really, how crazy is that?
"Here is this equation that describes this quantum phenomena... and when we follow the math to its conclusion - it tells us that positively charged electrons should exist. And they do exist!"

Amazing. 

[RoGeorge]

Indeed, one can discover by observing mathematical results.  It is even possible to make discoveries by crunching numbers (by simulation).  And in the recent years there are more and more papers with discoveries made by AI exploration.

What bothers me is that they all fail to explain why something happens.
Would be great to have a new type of mathematics, with embedded causality in it.

[Nominal Animal]

What bothers me is that they all fail to explain why something happens.

That is because we can only describe how, not why, in any language.  Even physics don't tell you why, only how.

I suspect 'why' is outside the scope, and belongs to philosophy or religion; one of those axiomatic things that are unprovable from the results/observations, which is all we have.

[RoGeorge]

The kind of "why" I am missing is not the philosophical one.  Only seeking to explain the sequence of events, what causes something to happen, or to manifest itself.

By explaining, I mean identifying the irreducible basic components that are needed for something to happen.  The choice of what would be considered as basic components is not important.  Can be the binary 0/1 from information theory, or the water/fire/air/earth, or any other choice, as long as they can be consistent along all the observations we can make in our physical world.

The idea of irreducible laws of physics (or infinite complexity), or the idea that time doesn't exist, seem more like theoretical speculation.  I see structure and causality everywhere, plus some progress with physics unification. 

[IanB]

What bothers me is that they all fail to explain why something happens.

If you start out with a simple harmonic system like the mass and spring with damping, then write down the equation that describes it from first principles, you will find that for certain parameter values the solution to the equation contains a term like sin(wt), meaning that it will oscillate. This sequence of steps from analysis to solution explains exactly why it oscillates in the best manner you will obtain.

If you want an intuition about the solution, you can consider what happens if you pull the mass away from the resting position and let it go. If, when the mass returns to the resting position, it is still moving, then it will oscillate. If, when the mass returns to the resting position, it has stopped moving, then it will not oscillate.

Would be great to have a new type of mathematics, with embedded causality in it.

There is already such a type of mathematics. It is called physics.

[Nominal Animal]

By explaining, I mean identifying the irreducible basic components that are needed for something to happen.

Righty-o.

One problem is that most (all?) things can be described in more than one way.  Consider optics and Snell's Law, versus Fermat's Principle of Least Time: the former can be derived from the latter, making them equivalent.  The two have very different 'explanations', too – even when they are mathematically equivalent.  Ouch.

Second problem is that we cannot always even tell whether our mathematical description is correct, or only approximately correct to within some arbitrary decision within the region we can measure.  An example of this is how Newtonian mechanics are indistinguishable from relativistic mechanics for small enough energies (masses and velocities).

Just like in mathematics, you always have some axioms that cannot be derived or proven in physics too.  Physics is about modeling or describing reality, not defining it.  (Thus, "physical law" = description or model that produces reproducible results matching real world experiments, not a dictat or requirement.)
 _ _ _ _ _

Of course, when we consider say usable oscillators, the problem is not a show-stopper.  We just need to outline what an "usable oscillator" is, and then analyse and group them so that we find their commonalities.  The simplest set I can find is the division into discrete states or infinitely differentiable amplitude ones, with most "real" oscillators a combination of the two; and this is what my answers above tried to show.

Unfortunately, there are many valid answers (groupings and derived commonalities/requirements).  Fortunately, it does not matter, because we can go back to the outline or definition of "usable oscillator", vary the definition or use case a bit, and pick the answer that yields the most useful properties.
Indeed, it actually is not unfortunate at all, because the set of different valid answers covers a larger set than any single answer alone; it just depends on the context and viewpoint which one happens to be most useful with respect to a given situation/problem/case.

If you think that is cheating or not true, do remember that even your own senses are approximations, inferences, and extrapolations.  We don't normally use infinite precision arithmetic in microcontrollers, because it would be a waste of resources.  Whenever investigating large molecules or systems of molecules, we don't use quantum mechanics derived ab initio models (because it would be too slow!), and use approximate/empirical ("descriptive") potential models instead.  We don't use relativistic mechanics for satellite orbits, because Newtonian mechanics gives the same results with much less effort, and the noise (interaction with solar wind, upper atmosphere, and magnetosphere, among other things) causes larger unpredictable errors than the difference between the two.  Perfect is the enemy of good: approximations and even guesses lets us advance, when requiring exact correctness would stall us.

In a very real way, even the 'irreducible basic components needed' kind of 'why' is almost impossible to answer, at least at this point.  Even if we had the language to describe it, we don't have the context in which it would make sense, be understandable to us humans.  It is annoying, but we are very imperfect beings.  We have to settle for the best approximation we have for it, and make do.
Remember, modern physics is barely a century old.

Anyone claiming "most of physics has been discovered already", is just repeating a variant of the "640k of RAM is enough for everyone".

Or heck, what do I know; I could be utterly wrong here.   Just trying to help you find a useful point of view and understanding.

[TimFox]

Indeed, one can discover by observing mathematical results.  It is even possible to make discoveries by crunching numbers (by simulation).  And in the recent years there are more and more papers with discoveries made by AI exploration.

What bothers me is that they all fail to explain why something happens.
Would be great to have a new type of mathematics, with embedded causality in it.

Even though most occurrences have a cause, they do not need to have a reason.

[jwet]

Nominal- There is a philosophy that teaches that there is only one ultimate truth but an infinite number of ways to observe it.  The older I get, the more I believe this I think, though I'm no closer to knowing the ultimate truth.  If this idea intrigues you, check out an interesting book - "The Surrender Experiment" by Michael Singer- it takes this to a very useful result.

One of the funny and useful things about online forums is that you see this at work- everyone brings their own truth to their understanding of the poster's problem.

I enjoy this thread- kind of a diversion.

[CatalinaWOW]

Nominal- There is a philosophy that teaches that there is only one ultimate truth but an infinite number of ways to observe it.  The older I get, the more I believe this I think, though I'm no closer to knowing the ultimate truth.  If this idea intrigues you, check out an interesting book - "The Surrender Experiment" by Michael Singer- it takes this to a very useful result.

One of the funny and useful things about online forums is that you see this at work- everyone brings their own truth to their understanding of the poster's problem.

I enjoy this thread- kind of a diversion.

.

This post also hints at one of the qualities of a good teacher.  The very best can identify the student's truth and provide explanations that are likely to be effective from that point of view, and ideally expanding understanding.

[aneevuser]

From a mathematical and physics point of view, you need to have a second order time derivative in the differential equation that describes the system. This is the only way the solution will include a complex exponential which will involve functions of sin() and cos().

Interesting thread.

The quote above is too restrictive though. There are higher order DEs which can be proven to oscillate, though I'm not at all familiar with the methods (and as you note later, coupled first order DEs like Lotka-Volterra can also oscillate - do we count them as "really" second order?) - there is a branch of DE maths called oscillation theory which treats these problems.