Abstract or Physical?

[quaternion]

I suppose a few comments on elliptic functions are in order. The NIST has extensive resources on elliptic functions available on their website: DLMF: 22 Jacobian Elliptic Functions

Section 22.19 (DLMF: 22.19 Physical Applications) lists physical applications. However, it will be readily seen that cases yielding closed form solutions are very limited, and are quite often chaotic. I hope I've made it clear my solution explicitly seeks non-chaotic solutions with a potential for general application.

Of all the applications of elliptic functions, the pendulum is the best known. Dozens, if not hundreds of references can be found on pendulums. The one I will note here is Bevivino, who carefully lays out the many different types of pendulum problems. Of all these, only the special case of the simple pendulum is strictly periodic & non-chaotic. For that one case there is no need to redefine time. [edit] (Given the scope I've set for this discussion. Of course there are numerous examples outside the scope of this thread where it is not necessary to redefine time.)

There may be other examples where that is true, but I am confident they remain applicable to only special cases.

With respect to the case I laid out for general dynamic systems, I clearly stated the model of time used in traditional vibration problems. I clearly stated the model of time used in my proposed solution. The two are different. If that is not a redefinition, I don't know what is.

To be perfectly transparent, what exactly is being redefined here is somewhat arbitrary. I have my reasons for saying it is time that is being redefined, but I'm open to hearing arguments it is something else. My reason is not metaphysical, but I'm even open to metaphysical arguments.

 

[FrumiousBandersnatch]

I suppose a few comments on elliptic functions are in order. The NIST has extensive resources on elliptic functions available on their website: DLMF: 22 Jacobian Elliptic Functions

Section 22.19 (DLMF: 22.19 Physical Applications) lists physical applications. However, it will be readily seen that cases yielding closed form solutions are very limited, and are quite often chaotic. I hope I've made it clear my solution explicitly seeks non-chaotic solutions with a potential for general application.

Of all the applications of elliptic functions, the pendulum is the best known. Dozens, if not hundreds of references can be found on pendulums. The one I will note here is Bevivino, who carefully lays out the many different types of pendulum problems. Of all these, only the special case of the simple pendulum is strictly periodic & non-chaotic. For that one case there is no need to redefine time. [edit] (Given the scope I've set for this discussion. Of course there are numerous examples outside the scope of this thread where it is not necessary to redefine time.)

There may be other examples where that is true, but I am confident they remain applicable to only special cases.

With respect to the case I laid out for general dynamic systems, I clearly stated the model of time used in traditional vibration problems. I clearly stated the model of time used in my proposed solution. The two are different. If that is not a redefinition, I don't know what is.

To be perfectly transparent, what exactly is being redefined here is somewhat arbitrary. I have my reasons for saying it is time that is being redefined, but I'm open to hearing arguments it is something else. My reason is not metaphysical, but I'm even open to metaphysical arguments.

I'm afraid I don't see it. Getting different results for the time taken when using different calculation methods isn't redefining time any more than getting different distances for a car journey when comparing odometer with GPS is redefining length.

 

[quaternion]

I'm afraid I don't see it. Getting different results for the time taken when using different calculation methods isn't redefining time any more than getting different distances for a car journey when comparing odometer with GPS is redefining length.

That's fine. Make your case.

To help, I'll back up to something simpler. If I integrate force with respect to time I get impulse, which is equivalent to a change in momentum. If I integrate force with respect to distance, I get work, which is equivalent to a change in energy. Momentum and energy are different things. If I change the variable of integration, we have two different things.

Do you disagree?

 

[FrumiousBandersnatch]

If I integrate force with respect to time I get impulse, which is equivalent to a change in momentum. If I integrate force with respect to distance, I get work, which is equivalent to a change in energy. Momentum and energy are different things. If I change the variable of integration, we have two different things.

Do you disagree?

No, that's reasonable.

 

[quaternion]

No, that's reasonable.

In order to get the solution I did, I changed the variable of integration from traditional problems like the pendulum problem. Something is different. My opinion is that the model of time is different.

If needed, I can keep going.

 

[sjastro]

I suppose a few comments on elliptic functions are in order. The NIST has extensive resources on elliptic functions available on their website: DLMF: 22 Jacobian Elliptic Functions

Section 22.19 (DLMF: 22.19 Physical Applications) lists physical applications. However, it will be readily seen that cases yielding closed form solutions are very limited, and are quite often chaotic. I hope I've made it clear my solution explicitly seeks non-chaotic solutions with a potential for general application.

Of all the applications of elliptic functions, the pendulum is the best known. Dozens, if not hundreds of references can be found on pendulums. The one I will note here is Bevivino, who carefully lays out the many different types of pendulum problems. Of all these, only the special case of the simple pendulum is strictly periodic & non-chaotic. For that one case there is no need to redefine time. [edit] (Given the scope I've set for this discussion. Of course there are numerous examples outside the scope of this thread where it is not necessary to redefine time.)

There may be other examples where that is true, but I am confident they remain applicable to only special cases.

With respect to the case I laid out for general dynamic systems, I clearly stated the model of time used in traditional vibration problems. I clearly stated the model of time used in my proposed solution. The two are different. If that is not a redefinition, I don't know what is.

To be perfectly transparent, what exactly is being redefined here is somewhat arbitrary. I have my reasons for saying it is time that is being redefined, but I'm open to hearing arguments it is something else. My reason is not metaphysical, but I'm even open to metaphysical arguments.

The reason I selected the simple pendulum as an example as it is representative of a device that keeps time, like a quartz watch or an atomic clock.
The common factor in each case is the dynamics is non-chaotic.

The Bevivino link shows that a varying external driving force will cause a simple pendulum to become chaotic; remove the external driving force and the pendulum is no longer chaotic and can keep time.

An interesting question that can be taken out of the link is what happens if external forces are applied to atomic clocks; will the atomic clocks become chaotic?
In fact we can refer to a 1960s experiment using radioactive Fe⁵⁷ nuclei for the answer.
Fe⁵⁷ nuclei have the interesting property (Mossbauser effect) of emitting and absorbing photons without the nuclei recoiling resulting in the photons being within a narrow frequency interval.
The experiment involves placing Fe⁵⁷ nuclei as an emitter at the centre of a rotor that can be operated at 30,000 RPMs to give accelerations 65,000X greater than gravity and Fe⁵⁷ nuclei on the perimeter of the rotor as the absorber.
The experiment can be analysed as the inertial frame of the emitter using special relativity or as the accelerated frame of the absorber using general relativity.
When the rotor is not operating the photons emitted from the centre are absorbed at the perimeter.
At 30,000 RPM absorption is largely reduced and is due to a change in frequency of the photons.
The change is not due to chaos; in the inertial frame the photon’s frequency change is due to the transverse Doppler effect or gravitational time dilation in the accelerated frame.

While time has “changed” due to the change in frequency it a successful prediction of both special and general relativity and doesn’t require a redefinition of time, as time in a rest frame is proper time which is invariant.

On a grander scale atomic clocks on GPS satellites operate at 10.23 MHz, on Earth the clocks are adjusted to run at 10.22999999543 MHz to compensate for special and general relativity effects and not a redefinition of time.

 

[sjastro]

A correction to the rotor experiment in my previous post.
The experimenters used a Co⁵⁷ emitter instead of Fe⁵⁷.

ABSTRACT
Using an ultracentrifuge rotor, the shift of the 14.4-keV Mössbauer absorption line of Fe⁵⁷ in a rotating system was measured as a function of the angular velocity ω. An Fe⁵⁷ absorber was placed at a radius of 9.3 cm from the axis of the rotor. A Co⁵⁷ source was mounted on a piezoelectric transducer at the center of the rotor. By applying a triangularly varying voltage to the transducer, the source could be moved relative to the absorber. This arrangement makes possible the observation of the entire resonance line at various values of ω. The measured transverse Doppler shift agrees within an experimental error of 1.1% with the predictions of the theory of relativity. Possible sources of systematic errors are discussed.

• Received 15 October 1962

DOI:Phys. Rev. 129, 2371 (1963) - Measurement of the Transverse Doppler Effect in an Accelerated System

©1963 American Physical Society