Author Topic: What's the minimum (physics first) to get an oscillator?  (Read 9538 times)

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Offline RoGeorgeTopic starter

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What's the minimum (physics first) to get an oscillator?
« on: May 23, 2023, 01:00:33 pm »
As the question say, how to explain an oscillator, and what's the must have for an oscillator?

By physics-first, I mean explaining based on energy, causality, time domain and alike easy to grasp notions.  Going from physics to math would be OK, the other way around, not so much.

Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #1 on: May 23, 2023, 01:39:00 pm »
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().

From an intuition point of view, the second order derivative will result from some concept of momentum in the system. Without momentum you cannot achieve oscillation.

Momentum can be thought of as a form of energy storage due to velocity (call it "kinetic" energy). Oscillation is the result of energy cycling between storage as "kinetic" energy and storage as potential energy. If the losses in the system ("damping") are low enough, the transfer of energy back and forth between the "kinetic" and potential forms of storage can result in an underdamped or cyclic behavior.

To make an oscillator you set up a system with cyclic (periodic) behavior, and then you feed in just enough energy to compensate for the losses, so that the oscillation continues indefinitely.
« Last Edit: May 23, 2023, 01:50:43 pm by IanB »
 

Offline TimFox

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #2 on: May 23, 2023, 01:54:52 pm »
That is a reasonable description of an "harmonic oscillator".  Note that any such device needs an external power source to maintain the oscillation.
The prototype harmonic oscillator in physics discussions is a spring (force proportional to displacement) and a mass, where the position of the mass oscillates.
Another oscillator type is the "relaxation oscillator", where only first derivatives appear in the analysis.  A 555 or a neon bulb can provide the hysteresis that bounds the oscillation.
 

Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #3 on: May 23, 2023, 01:58:36 pm »
I think even a relaxation oscillator would include a second order time derivative in its description (albeit hidden and very nonlinear). This is because a system containing more than one first order derivative can exhibit second order behavior (the converse is true also: a second order ODE can be decomposed into a paired system of first order ODEs).
 

Offline CatalinaWOW

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #4 on: May 23, 2023, 02:16:42 pm »
I think even a relaxation oscillator would include a second order time derivative in its description (albeit hidden and very nonlinear). This is because a system containing more than one first order derivative can exhibit second order behavior (the converse is true also: a second order ODE can be decomposed into a paired system of first order ODEs).

The relaxation oscillator highlights the fuzzy line between ideal models and reality.  The conceptual oscillator can't be fully described by a differential equation because the trigger circuit is non continuous and has no meaningful derivatives.  But it can be approximated mathematically and the physical reality can be matched by those approximations.
 
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Offline T3sl4co1l

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #5 on: May 23, 2023, 02:20:35 pm »
Oscillator in the simplest physics sense (a passive oscillatory system e.g. spring-and-mass), or practically speaking (a driven system with feedback)?  Or more specifically still, a combination of the two (a periodic or even ~sinusoidal oscillator)?

A complete description to some very, very basic level, I think you'll have some trouble expressing, but if we say it's a given that atoms and molecules exist at around room temperature, then a basic chemical oscillator operating on a concentration gradient of reactants might be sufficient.

Not sure offhand if there are simpler systems than that; perhaps some gas molecules have interesting enough dynamics that say a heat flux, or light, or excess kinetic energy (a molecular or particle beam?) is enough to induce oscillation.  We could of course point to plasmas, but shall we include atomic transitions as instances of oscillation or not?

There are molecular vibration modes that closely resemble a spring-and-mass system (quantum harmonic oscillator, QHO), wherein the energy levels are quantized, but the transition energy between adjacent states is approximately constant -- equivalent to the fixed resonant frequency of the classical harmonic oscillator.  This is in contrast to the more sporadic modes of a bare atom for example, where similar dynamics might apply (with suitable stimulus (i.e., from its spectrum), we can increment between electron orbital levels), but not all states can be transitioned between, in a single event, due to spin selection rules.  Whereas the QHO has a "ladder" state that allows free exchange of potential energy at the same frequency.

From the more abstract side -- a linear ODE isn't sufficient description, as we necessarily have limited sources and sinks of energy/power.  We can get nearly perpetual oscillators if we remove all influences; a lone planet in circular orbit of a lone star, pretty closely obeys such an equation, over time scales of billions of years (for suitable definition of "lone"); but not for all time, and eventually there will be significant exchange of energy with surrounding systems.  (If nothing else, even for ideal rigid bodies orbiting in isolation from any others, gravitational waves will decay the orbit over heat-death time scales.  Or maybe it's less than that?  I forget.)  Particularly for a driven system, we must include a logistic function that, given an input, limits maximum amplitude or energy exchange; and extraction, given an output.  All real systems are nonlinear, having such a limit.

More generally, we might specify an oscillator with a limit cycle, given some assumed energy input, exchange, and output.  The 2nd order term might even be dominant (as in the above case), but other terms are necessary in a real system.

The complexity then, is finding a way to express these terms, which is both simple enough to be useful, and real enough to be meaningful.  I'm afraid I don't know enough about DEs anymore to help further with such a description.  Anyway, the plan would be to start from a high level like this, and also from a low level (in whatever medium you like, be it fields, particles or combinations thereof), and meet in the middle with a complete description of an implementation/realization of that system.

Assuming I've interpreted your meaning and intent correctly, of course!

Tim
« Last Edit: May 23, 2023, 02:24:14 pm by T3sl4co1l »
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Offline rstofer

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #6 on: May 23, 2023, 03:20:53 pm »
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().

From an intuition point of view, the second order derivative will result from some concept of momentum in the system. Without momentum you cannot achieve oscillation.

Momentum can be thought of as a form of energy storage due to velocity (call it "kinetic" energy). Oscillation is the result of energy cycling between storage as "kinetic" energy and storage as potential energy. If the losses in the system ("damping") are low enough, the transfer of energy back and forth between the "kinetic" and potential forms of storage can result in an underdamped or cyclic behavior.

To make an oscillator you set up a system with cyclic (periodic) behavior, and then you feed in just enough energy to compensate for the losses, so that the oscillation continues indefinitely.

I really like this description.  I have spent way too much time on the mass-spring-damper problem and analog computing to let this description slide by.
 
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Offline CatalinaWOW

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #7 on: May 23, 2023, 05:27:34 pm »
The drinking bird toy is an example of a thermodynamic oscillator.
 
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Offline ejeffrey

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #8 on: May 23, 2023, 06:22:43 pm »
Mostly it depends on how you define an oscillator.

I generally consider an oscillator as something with positive feedback and gain greater than one, and thus an energy consuming processes.  That is, you feed it with power and it produces an oscillating signal.  This would include things like crystal oscillators, relaxation oscillators like a 555, drinking birds, lasers, and so on.

But that excludes entirely passive things like spring/mass systems or LC circuits, which I would call "resonators".  Resonator + gain + feedback => oscillator.  It also excludes simple spinning objects like a pulsar which aren't (according to my general usage) either a resonator or an oscillator, it's just something that is periodic.  This is consistent with the typical use in electronics where a bare quartz crystal is just called a "crystal", and a unit packaged with an amplifier is called a "crystal oscillator".

But that's mostly a question of terminology, and how I use it.  Other people in other contexts use those terms slightly differently.  In physics, the specific phrase"simple harmonics oscillator" means any system with a second order linear differential equation.  It can be driven, undriven, damped, or have the damping ignored.  If you sharpen your question enough to be exact, the answer will probably be obvious.
 

Offline westfw

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #9 on: May 23, 2023, 06:25:34 pm »
Quote
The prototype harmonic oscillator in physics discussions is a spring


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

 

Offline Nominal Animal

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #10 on: May 23, 2023, 06:34:27 pm »
Oscillator in the simplest physics sense (a passive oscillatory system e.g. spring-and-mass), or practically speaking (a driven system with feedback)?  Or more specifically still, a combination of the two (a periodic or even ~sinusoidal oscillator)?
And do we include more esoteric phenomena like quantum time crystals, where the lowest energy state is a non-static, repetitive one?

It all boils down to what you specifically and exactly mean by an oscillator.  In my opinion, one useful disctinction is between the phenomenon, and as an usable tool.
 _ _ _ _ _ _

For the tool view, any sufficiently regular repetitive system whose phase space (behaviour with respect to any variable that may affect it) is such that useful amounts of energy can be exchanged (injected and/or extracted) without destabilizing the system, can be used as an oscillator.

Basically all useful oscillators – all continuous-amplitude oscillators! – can be described as harmonic oscillators, damped harmonic oscillators, or driven harmonic oscillators.  Even non-harmonic oscillators can be described as damped-driven harmonic oscillators, where the damping-driving force depends on the phase of the oscillator; with the base harmonic oscillator being a near approximation, with the damping-driving force causing only relatively small changes to the period and/or amplitude.

The reasons for this are actually very interesting, and can be divided into two separate classes.

One class is that any convex shape is analogous to a sphere.  (There is a better mathematical term for this, but I can't recall it right now.)  In the oscillator case, because we are interested in repetitive state changes, we care only about the interval at which the repetition occurs.  So, whether nominal sub-states take an equal amount of time, does not matter.  In essence, we don't care that much about the overall "shape" of the oscillation.  (Insert automotive analogy about lap times with track of the same length but different shapes, or tracks of different length and shape but yielding same/similar lap times for each driver, here.)

The other class is how any periodic continuous signal, say the amplitude of any continuous-state oscillator, can be decomposed into an (infinite) set of sinusoidal signals, and how even non-continous signals can be approximated to any desired precision: Fourier series.  (Sine and cosine themselves being the solutions to any pure harmonic oscillator.)

There is a third one, too: the smaller the amplitude compared to the energies and distances involved in an oscillator, the closer it seems to match a harmonic oscillator.  However, I myself do not know why this is, or even if it is just a practical result of a noisy universe (i.e., any non-harmonic components drowned in noise in real life), so I consider it more or less just a rule of thumb.
 

Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #11 on: May 23, 2023, 06:40:03 pm »
The relaxation oscillator highlights the fuzzy line between ideal models and reality.  The conceptual oscillator can't be fully described by a differential equation because the trigger circuit is non continuous and has no meaningful derivatives.  But it can be approximated mathematically and the physical reality can be matched by those approximations.

This is an interesting question. What is an "ideal model"? It is presumably a model where simplifications have been made to eliminate some less significant details and make it easier to work with. For example, texts on differential equations nearly always introduce the student to linear differential equations because they often permit an analytical solution to be obtained. But this is an idealization, since the real world is not often linear.

In the real world beyond college and textbooks, complex systems are simulated numerically on a computer, and (possibly gross) nonlinearities in the system can be accommodated.

For example, suppose you put an audio amplifier into a feedback loop by putting the microphone close to the speaker. The amplifier will saturate, the signal will clip like crazy, and the whole thing will be horribly nonlinear. Yet the system is made up of analog components that can be described with differential equations. There is undoubtedly no analytical solution, but a computer simulation if programmed sufficiently accurately could reproduce in a model what the feedback loop does in real life.
 

Offline RoGeorgeTopic starter

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #12 on: May 23, 2023, 06:43:26 pm »
The question started from an LC oscillator.

At a first look, an oscillator would need an amplifier to substitute the lost energy, a limiter to keep the amplitude constant, and a positive feedback loop.  Those alone won't cut it, would only make a latch.  It needs some inertia/delay/memory (the LC tank in the starting example).

This is where I couldn't decide what was the broader concept to pick.  It can be a resonant oscillator as an LC, or it can be a forced oscillator with a comparator and something that flips the state, as in the drinking bird example.

It gets even more bamboozling when thinking about numerical oscillators.  What would mean energy conservation for a numerical oscillator, or momentum, or seeking minimum energy equilibrium, or inertia?  All these are very important and inescapable in a physical oscillator, yet they don't make much sense, and they seem totally arbitrary for a numerical oscillator.

That is why asking to identify the essential components of a physical oscillator, at first.
« Last Edit: May 23, 2023, 06:53:56 pm by RoGeorge »
 

Offline ejeffrey

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #13 on: May 23, 2023, 06:45:22 pm »
Quote
The prototype harmonic oscillator in physics discussions is a spring


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

No, at least in my description / definition above a simple pendulum isn't any different.  In that case, gravity is just providing the spring force.  Now a pendulum clock has a mechanism to detect the motion of the arm and provide kicks, powered by a falling weight.  That I would call an oscillator.
 

Offline Nominal Animal

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #14 on: May 23, 2023, 07:23:56 pm »
The question started from an LC oscillator.

At a first look, an oscillator would need an amplifier to substitute the lost energy, a limiter to keep the amplitude constant, and a positive feedback loop.  Those alone won't cut it, would only make a latch.  It needs some inertia/delay/memory (the LC tank in the starting example).
You end up with negative feedback.  Which gives you a harmonic oscillator.
When we include the "amplifier" and loss of energy, we get a damped-driven harmonic oscillator, where the damping is due to losses, and the driving matches the damping.  Depending on the damping and driving mechanisms, they can cause a small change in the period.

Note that while the amplitude of the harmonic oscillator depends on the energy involved, the period does not.  So, small perturbations or fluctuations in the amount of damping or driving does not affect the period of the oscillator, as long as the amplitude stays within the elastic/ergodic range (where the same amount of energy that was needed to cause the change is released when returning to the original position) and the change is smooth and slow enough.  Typically, the driving and damping are smooth and interdependent (matching each other intrinsically, both changing only slowly and smoothly, continuously, across multiple periods), so that their effect on the oscillator period is neglible, within environmental noise.

It gets even more bamboozling when thinking about numerical oscillators.
Numerical oscillators?  Do you mean periodic functions and periodic sequences?  They are based around periodicity, and from a physics perspective, they are "pure": more like empty universes than systems of interacting parts.
« Last Edit: May 23, 2023, 07:32:32 pm by Nominal Animal »
 

Offline nctnico

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #15 on: May 23, 2023, 07:25:34 pm »
That is why asking to identify the essential components of a physical oscillator, at first.
The way I see it is that in essence you'll need a 'body' that can resonate at a certain fundamental frequency. And then you'll need an energy source that provides energy to excitate the body to resonate at the fundamental frequency or a harmonic. The fundamental frequency is typically determined by how fast waves (electric, acoustic, pressure, etc) can travel through the body and bounce back from the edges.

Even sea waves are a form of oscillation.
« Last Edit: May 23, 2023, 07:31:30 pm by nctnico »
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Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #16 on: May 23, 2023, 08:12:22 pm »
That is why asking to identify the essential components of a physical oscillator, at first.

For a harmonic oscillator the essential component is that the mathematical description includes some kind of 2nd order differential equation in time. With a first order system harmonic oscillation cannot happen.

Another kind of oscillator results from switching, for example if you use logic elements to make a square wave. However, if you analyze at a small enough time scale even logic circuits are analog, and somewhere in the description will be 2nd order characteristics.
« Last Edit: May 23, 2023, 09:08:42 pm by IanB »
 

Offline TimFox

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #17 on: May 23, 2023, 09:06:56 pm »
Quote
The prototype harmonic oscillator in physics discussions is a spring


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

My full statement was "The prototype harmonic oscillator in physics discussions is a spring and a mass."
When you analyze the force on the mass (pendulum bob) due to gravity and the constraint of motion with a constant radius, you see a restoring force proportional to the sine of the pendulum angle, which for small amplitudes is extremely close to a force proportional to the angle itself.  This gives the same equation as a mass connected to a spring, where the restoring force is proportional to the displacement (and reverses sign with the sign of the displacement).

By the way, when you add "damping" to the simple harmonic oscillator, you need a damping force proportional to the velocity (also reversing with velocity).  In a mechanical system, this is conventionally a "dashpot", which is the same basic device used in an automotive shock absorber.  Friction does not have the ideal dependences.  Such a damping force is analogous to the electrical loss (resistance) in an RLC electrical oscillator.
 

Offline Nominal Animal

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #18 on: May 23, 2023, 09:22:26 pm »
The only oscillators you cannot describe/approximate (to any desired precision) as some form or combination of (ideal/damped/driven) harmonic oscillator(s) are those based on actual physically discrete states, like certain quantum phenomena.  Indeed, one second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 (133Cs).

The duration of many quantum phenomena is surprisingly stable; they're just so darned short they're difficult to investigate.

I'm not exactly sure what RoGeorge meant by numerical oscillators, but all practical use cases I can think of relies on either discrete states or discrete time variable steps, so they're better modeled as periodic systems with either discrete states, or discretely sampled states instead.

For example, in Conway's Game of Life, there are a number of periodic patterns, both static and moving.  In many ways, these can be considered time crystals: their base state is not a single state, but a sequence of discrete states, that proceeds without interaction or any transfer of energy.
 

Offline CatalinaWOW

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #19 on: May 24, 2023, 01:38:01 am »
The relaxation oscillator highlights the fuzzy line between ideal models and reality.  The conceptual oscillator can't be fully described by a differential equation because the trigger circuit is non continuous and has no meaningful derivatives.  But it can be approximated mathematically and the physical reality can be matched by those approximations.

This is an interesting question. What is an "ideal model"? It is presumably a model where simplifications have been made to eliminate some less significant details and make it easier to work with. For example, texts on differential equations nearly always introduce the student to linear differential equations because they often permit an analytical solution to be obtained. But this is an idealization, since the real world is not often linear.

In the real world beyond college and textbooks, complex systems are simulated numerically on a computer, and (possibly gross) nonlinearities in the system can be accommodated.

For example, suppose you put an audio amplifier into a feedback loop by putting the microphone close to the speaker. The amplifier will saturate, the signal will clip like crazy, and the whole thing will be horribly nonlinear. Yet the system is made up of analog components that can be described with differential equations. There is undoubtedly no analytical solution, but a computer simulation if programmed sufficiently accurately could reproduce in a model what the feedback loop does in real life.

There is a language problem here.  Ideal vs idealized.  Ideal in the way you used it means best, or optimum or most useful.  Which in different situations can mean simplest or most accurate or cheapest to execute.  Idealized in the sense I meant is more in the sense of prototypical.  An idealized square wave has zero rise time.  And can be approximated to any desires accuracy by a series of harmonic oscillators.  But never exactly recreated.  So technically this square wave isn't the solution of a differential equation.   But I am an engineer and the harmonic series are close enough.
 

Offline rstofer

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #20 on: May 24, 2023, 01:39:44 am »
Attached is a MATLAB/Simulink simulation of a mass-spring-damper.  If the D terms goes to 0, it is an oscillator.

The solution is based on the work of Lord Kelvin.  Basically, we assume we have y'' (but we don't yet) and integrate it to get y' and then integrate again to get y.  Now we sum up the scaled values to create y'' and stuff it back in the input (now that we actually have it).  This is the genius of analog computing; integrate until you get rid of the derivatives.  Notice the IC=-1 term adjacent to the second integrator.  It sets the initial condition to -1.

But, as stated above, you need a second order equation to get oscillation.


« Last Edit: May 24, 2023, 01:47:22 am by rstofer »
 

Offline thermistor-guy

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #21 on: May 24, 2023, 03:38:17 am »
As the question say, how to explain an oscillator, and what's the must have for an oscillator?
...

Oscillation in what domain, exactly?

Predator-prey populations can oscillate.

Many organisms have a sleep-wake cycle, and this is old in evolutionary terms:
  https://www.caltech.edu/about/news/surprising-ancient-behavior-jellyfish-79701

There is a long cycle in human affairs. We are currently in the cycle phase called "The Fourth Turning"
  https://en.wikipedia.org/wiki/Strauss%E2%80%93Howe_generational_theory

When you say "physics first", I assume you are reaching for something fundamental to all
domains where oscillation occurs. Are you aiming the question at electrical and mechanical
systems only?
 

Offline Brumby

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #22 on: May 24, 2023, 03:46:10 am »
My first thought was: Build an amplifier.   ;D
 
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Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #23 on: May 24, 2023, 04:09:07 am »
There is a language problem here.  Ideal vs idealized.  Ideal in the way you used it means best, or optimum or most useful.  Which in different situations can mean simplest or most accurate or cheapest to execute.  Idealized in the sense I meant is more in the sense of prototypical.  An idealized square wave has zero rise time.  And can be approximated to any desires accuracy by a series of harmonic oscillators.  But never exactly recreated.  So technically this square wave isn't the solution of a differential equation.   But I am an engineer and the harmonic series are close enough.

I was using ideal in exactly the sense you described, to mean idealized or perfect. So I would say that an ideal or perfect square wave model has been simplified to remove such imperfections as finite rise time, or overshoot, that inevitably occur in the real world and that do not need to be considered for many purposes.
 

Online IanB

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Re: What's the minimum (physics first) to get an oscillator?
« Reply #24 on: May 24, 2023, 04:15:46 am »
Oscillation in what domain, exactly?

It doesn't really matter, due the principle of analogous systems. It turns out that if you construct a mathematical model starting in any given domain, and then reduce it to dimensionless form, you generally end up with the same model to describe the same kind of system behavior.
 


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