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December 29

Potential energy vs. kinetic energy. Why not also "potential velocity" vs. "kinetic velocity"? E.g. in the following case:

In a harmonic oscillator, reaching the highest point involves - both a minimal kinetic energy - along with a maximal potential energy, whereas reaching the lowest point involves - both a maximal kinetic energy - along with a minimal potential energy. Thus the mechanical energy becomes the sum of kinetic energy + potential energy, and is a conserved quantity.

So I wonder if it's reasonable to define also "potential velocity" vs. "kinetic velocity", and claim that in a harmonic oscillator, reaching the highest point involves - both a minimal "kinetic velocity" (i.e. involves what we usually call a rest) - along with a maximal "potential velocity", whereas reaching the lowest point involves - both a maximal "kinetic velocity" (i.e. involves what we usually call the actual velocity) - along with a minimal "potential velocity". Thus we can also define "mechanical velocity" as the sum of "kinetic velocity" + "potential velocity", and claim that the mechanical velocity is a conserved quantity - at least as far as a harmonic oscillator is concerned.

Reasonable?

Note that I could also ask an analogous question - as to the concept of "potential momentum", but this term is already used in the theory of hidden momentum for another meaning, so for the time being I'm focusing on velocity.

HOTmag (talk) 12:26, 29 December 2024 (UTC)

'kinetic velocity' is just 'velocity'. 'potential velocity' has no meaning. Andy Dingley (talk) 13:56, 29 December 2024 (UTC)

Per my suggestion, the ratio between distance and time is not called "velocity" but rather "kinetic velocity".

Further, per my suggestion, if you don't indicate whether the "velocity" you're talking about is a "kinetic velocity" or a "potential velocity" or a "mechanical velocity", the very concept of "velocity" alone has no meaning!

On the other hand, "potential velocity" is defined as the difference between the "mechanical velocity" and the "kinetic velocity"! Just as, this is the case if we replace "velocity" by "energy". For more details, see the example above, about the harmonic oscillator. HOTmag (talk) 15:14, 29 December 2024 (UTC)

You could define the potential velocity of a body at a particular height as the velocity it would hit the ground at if dropped from that height. But the sum of the potential and kinetic velocities would not be conserved; rather $v_{\mathrm{t}\mathrm{o}\mathrm{t}}=\sqrt{v_p^2+v_k^2}$ would be constant. catslash (talk) 18:54, 29 December 2024 (UTC)

Thank you. HOTmag (talk) 20:07, 29 December 2024 (UTC)

'Potential velocity' has no meaning. You seem to be arguing that in a system where energy is conserved, but is transforming between kinetic and potential energy, (You might also want to compare this to conservation of momentum.) then you can express that instead through a new conservation law based on velocity. But this doesn't work. There's no relation between velocity and potential energy.

In a harmonic oscillator, the potential energy is typically coming from some central restoring force with a relationship to position, nothing at all to do with velocity. Where some axiomatic external rule (such as Hooke's Law applying, because the system is a mass on a spring) happens to relate the position and velocity through a suitable relation, then the system will then (and only then) behave as a harmonic oscillator. But a different system (swap the spring for a dashpot) doesn't have this, thus won't oscillate. Andy Dingley (talk) 00:00, 30 December 2024 (UTC)

Let me quote a sentence from my original post: Template:Tq

What's wrong in this quotation? HOTmag (talk) 07:52, 30 December 2024 (UTC)

It is true, not only for harmonic oscillators, provided that you define Template:Math.  --Lambiam 09:07, 30 December 2024 (UTC)

• You have defined some arbitrary values for new 'velocities', where their only definition is that they then demonstrate some new conservation law. Which is really the conservation of energy, but you're refusing to use that term for some reason.

As Catslash pointed out, the conserved quantity here is proportional to the square of velocity, so your conservation equation has to include that. It's simply wrong that any linear function of velocity would be conserved here. Not merely we can't prove that, but we can prove (the sum of the squares diverges from the sum) that it's actually contradicted. For any definition of 'another velocity' which is a linear function of velocity.

Lambiam's definition isn't a conservation law, it's merely a mathematical identity. The sum of any value and its additive inverse is always zero. Andy Dingley (talk) 14:04, 30 December 2024 (UTC)

Template:Small  --Lambiam 11:20, 31 December 2024 (UTC)

We have a perfectly viable definition of potential energy. For a pendulum it's based on the change in height of the pendulum bob against gravity. For some other oscillators it would involve the work done against a spring. Andy Dingley (talk) 16:33, 31 December 2024 (UTC)

Oops, I mistyped. I meant to write:

"Template:Small"

 --Lambiam 23:32, 31 December 2024 (UTC)