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Kartazion

Anharmonic Oscillator

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Hello,

Here is the equation of an anharmonic oscillator; For example, a mass m fixed to a special spring is considered here:

$$m \frac{d^2 x}{dt^2}- K x+AKx^3=0$$

I want to modify this equation to insert 50 cycles of the round trip of the mass in one second.
How to write this sequel? (I do not know where and what value to modify)

Thank you

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That’s the equation, but it’s not the solution, which would give you x(t)

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Have you tried exploring the following sequence


[math]m\frac{{{d^2}x}}{{d{t^2}}} = F\left( x \right) = Kx\left( {1 - A{x^2}} \right)[/math]


[math]\frac{{dx}}{{dt}} = [/math]


[math]t = [/math]

or otherwise separating the variables?

Edited by studiot

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Thank you for your answers.

I will try to exploit the sequence of studiot, and the x(t) of swansont.

However I saw on the web that there was no analytical solution to the problem (as swansont said) and that it was necessary, in some cases, to make a numerical integration of the Runge Kutta methods type to order 4. :(
What can this Runge Kutta method do for this oscillator?

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Good evening

What I wish is to be able to submit a cycle reiteration to the equation. I consider a cycle, to a complete round-trip of the mass m on its axis.

If I multiply the execution of the equation to one million cycles per second, then the mass density, visually speaking, would be distributed at the ends. Indeed, the behavior of the oscillator determines a slowdown, then a certain stop, on each end position of the mass m.

anharmonic-oscillator.png

Do you validate this deduction?

 

Edited by Kartazion

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17 hours ago, Kartazion said:

Good evening

What I wish is to be able to submit a cycle reiteration to the equation. I consider a cycle, to a complete round-trip of the mass m on its axis.

If I multiply the execution of the equation to one million cycles per second, then the mass density, visually speaking, would be distributed at the ends. Indeed, the behavior of the oscillator determines a slowdown, then a certain stop, on each end position of the mass m.

anharmonic-oscillator.png

Do you validate this deduction?

 

Probably, but you would have to solve the equation to be sure. Note that you can't arbitrarily change the frequency of the oscillation. That typically comes from the solution to the problem. If you specify it, then some other quantity has to be unknown and free to change. You would not be comparing identical systems. 

Using a simple example, a pendulum's frequency depends on its length, and g. If you dictate that it must have a certain frequency, you have to change the length and/or find a place where g has the proper value to get this result. If it naturally swings at 1 Hz, it's not going to magically swing at 1 MHz

 

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4 hours ago, swansont said:

Using a simple example, a pendulum's frequency depends on its length, and g. If you dictate that it must have a certain frequency, you have to change the length and/or find a place where g has the proper value to get this result. If it naturally swings at 1 Hz, it's not going to magically swing at 1 MHz

That's why I thought of a field that directs the particle rather than a spring or wire. A directing field which oscillates the particle from a point A to a point B at very high frequency.
Are there no calculations on this subject? Duplication of mass density according to its speed of movement in space?

mass-asleep.png

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One way to handle this situation is to 'assume' a potential function as a solution, expanded as a Taylor series.

I will post some more later.

Going to a lecture on JC Maxwell tonight by Brian Clegg.

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OK so here is more.

Just keep in mind that swansont has repeated told you you need to have an idea of the form of the solution.
 

Here is the energy (potential ) approach.
The first three slides show a simple anharmonic oscillator (there are many types).
Not the words on slide 1 that the potential well is symmetric for a harmonic oscillator and asymmetric for an anharmonic one as shown in their diagram 8.9.

The fourth slide shows types of potentials in fig 17.1. The third one is the one discussed in the other book.

Anharmonic equations can lead to elliptic integrals whcih have no analytical solution, so yes, these need tables or other numerical methods.

anhar1.thumb.jpg.772b7a453cf905caf54bb3ac8f8e73d2.jpganhar2.thumb.jpg.8706fbb82b604150660140e73285479b.jpganhar3.thumb.jpg.42327526bb11ebee52eae4d55cbb6a58.jpg

 

anhar4.thumb.jpg.4a927415a7892c63def44f1512ce8544.jpg

Edited by studiot

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Going back to your original equation, please say what is your interest in this.
You seem very specific in you questions/requirements, but rather woolly in your understanding of what the equation represents.

Is this homework whre you are trying to answer a specific question?

Or is this general study of anharmonic oscillators or what?

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44 minutes ago, studiot said:

Going back to your original equation, please say what is your interest in this.

The equation corresponds to the desired movement of the particle.

 

44 minutes ago, studiot said:

You seem very specific in you questions/requirements, but rather woolly in your understanding of what the equation represents.

Yes I do not understand the equation, but I saw its simulation at this address in French (you have to click on "départ" to be able to see the yellow point to move).

 

44 minutes ago, studiot said:

Is this homework whre you are trying to answer a specific question?

Or is this general study of anharmonic oscillators or what?

No, it's not Homework. I want to prove mathematically that to alternate at the speed of light a particle at a point A to a point B, would be seen as two fixed and static points. I just want to accelerate the movement of the particle back and forth so that there is a split of the mass. Indeed it allows me eventually to be able to distribute the mass in several separate places.

matrix_1_courbe.png


 

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On 11/23/2019 at 4:13 PM, Kartazion said:

The equation corresponds to the desired movement of the particle.

 

Yes I do not understand the equation, but I saw its simulation at this address in French (you have to click on "départ" to be able to see the yellow point to move).

 

No, it's not Homework. I want to prove mathematically that to alternate at the speed of light a particle at a point A to a point B, would be seen as two fixed and static points. I just want to accelerate the movement of the particle back and forth so that there is a split of the mass. Indeed it allows me eventually to be able to distribute the mass in several separate places.

matrix_1_courbe.png


 

I see, thank you.

You have good English so why did you not say all this at the beginning?

There are several things you still haven't mentioned such as where did you get this eqaution and why are you not using a simple harmonic oscillator.

You say your interest is in calculating a velocity but you haven't indicated your level in Mathematics (or Physics) although you have posted this in the Mathematics section, not the Physics one.

So  here (assuming you know what a differential equation is) is some mathematics.

Take your basic equation


[math]m\frac{{{d^2}x}}{{d{t^2}}} - Kx + AK{x^3} = 0[/math]

Make these substitutions for the constants m, A and K


[math]K =  - am[/math]


[math]AK = bm[/math]

Then the equation becomes


[math]\frac{{{d^2}x}}{{d{t^2}}} + ax + b{x^3} = 0[/math]


Rearrange and multiply through by     [math]2\frac{{dx}}{{dt}}[/math]


[math]2\frac{{dx}}{{dt}}\frac{{{d^2}x}}{{d{t^2}}} = 2\frac{{dx}}{{dt}}\left( {ax - b{x^3}} \right)[/math]


Integrate


[math]{\left( {\frac{{dx}}{{dt}}} \right)^2} = C - a{x^2} - \frac{1}{2}b{x^4}[/math]

and note that


[math]\frac{{dx}}{{dt}} = {\rm{velocity}}[/math]

 

 

You do not have to solve the equation to find the velocity as shown above.
You can also find the velocity by Physics arguments, noting that my rearrangement is basically formulating Newton's second law and that the velocity is the integral of the acceleration and at a max when the acceleration is zero.

 

Your equation itself is knows as Duffing's Equation, which is why I wondered why you chose it.

This is a non linear equation that appears in several guises, not the least Chaos theory where another form of solution is used by plotting the complex phase plane response.
Further forms of solution using Fourier series are also possible and used in Mechanics considering masses vibrating against special non linear springs.

It is a good equation to discuss but I think you will have difficulty applying it to Special Relativity.

 

Edited by studiot

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Thank you for all this, it's really interesting.

 

On 11/25/2019 at 10:39 PM, studiot said:

You say your interest is in calculating a velocity but you haven't indicated your level in Mathematics (or Physics) although you have posted this in the Mathematics section, not the Physics one.

My level in math is rather passable; but I'm starting to understand. Give me a little more time to assimilate. It is the mathematical equation that interests me.

 

On 11/25/2019 at 10:39 PM, studiot said:

This is a non linear equation that appears in several guises, ...

I do not know if there is a report, but I wish to spatialize the mass m. I would like to know how to answer in equation, but I can explain it this way:

The following example would be this time of alternating the mass m between three different points (three different ends).
So far there were only two, A and B (maximum left end of m on the x axis and maximum right end of m on the x axis).

 

anharmonic-oscillator-4.png

 

From position A, the mass m work in rectilinear motion along x until it reaches the extreme position of B. Then m returns to the postion of A.
We can add from position A, a new path of m on the axis x' to B'.

B' being a position different from B.

 

anharmonic-oscillator-5.png

In fact the position A at its center gives by rotation the radius of x and the path from m to B or B'. This angular rotation is noted θ for example.
If I add some degree θ to A on m, then the projection of the radius will be of a different axis.
In my case it amounts to alternating the degrees θ, in addition to the positions A, B and B'.

I'll just want to write that like that:

(A-->B-->A-->B'-->) = one complete cycle

(A-->B-->A-->B'-->) + (A-->B-->A-->B'-->) = two complete cycles

 

But I think the equation must start with:

$$ \theta \frac{dx}{dt} $$

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13 hours ago, Kartazion said:

My level in math is rather passable; but I'm starting to understand. Give me a little more time to assimilate. It is the mathematical equation that interests me

 

Mathematical equations may or may not correspond to physical situations.

Unfortunately they seem rather disconnected in this thread.

So I am at a loss to know where you are trying to go with this.

I will wait to hear the result of your 'assimilation' before commenting further.

Edited by studiot

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On 11/23/2019 at 5:13 PM, Kartazion said:

 I want to prove mathematically that to alternate at the speed of light a particle at a point A to a point B, would be seen as two fixed and static points. I just want to accelerate the movement of the particle back and forth so that there is a split of the mass. Indeed it allows me eventually to be able to distribute the mass in several separate places.

Several problems with this.

1. Objects with mass cannot move at the sped of light.

2. Even if we allowed it to travel at the speed of light, it would not be seen as two fixed points; it would appear as an object moving at the speed of light between those two points. It would be at point A and then at a non-zero time later, it would be at point B. The time between it being at A and B would depend on how far apart they are.

3. The mass would never be at different places at the same time, so I don't think the phrase "distribute the mass in several separate places" really means anything. The mass is no more "distributed in several separate places" than my mass is when I drive to work and back.

(That is the practical aspect. Can't help with the math, as I don't understand what you are trying to do.)

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1 hour ago, studiot said:

Mathematical equations may or may not correspond to physical situations.

Unfortunately they seem rather disconnected in this thread.

So I am at a loss to know where you are trying to go with this.

Yes, I wish in the long term (much later) to explode quantum states of type supperposition, or entanglement.
I even suspect that the provence of quantum chromodynamics has originated in the oscillation of a single moving particle.

But this is for later.

chromodynamic-colors.png

vide-quantique.png

 

On 11/25/2019 at 10:39 PM, studiot said:

Integrate

{\left( {\frac{{dx}}{{dt}}} \right)^2} = C - a{x^2} - \frac{1}{2}b{x^4}

 
C is
the speed of light?

 

1 hour ago, Strange said:

1. Objects with mass cannot move at the sped of light.

The velocity of quarks within the nucleus is nearly equal to that of light.
The quarks, which are the components of protons and neutrons, move back and forth at a speed close to the speed of light, and in random directions. This back and forth movement, or zigzag motion, has already been quantified [6] .

https://file.scirp.org/Html/12-7502384_60688.htm

 

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1 hour ago, Strange said:

2. Even if we allowed it to travel at the speed of light, it would not be seen as two fixed points; it would appear as an object moving at the speed of light between those two points. It would be at point A and then at a non-zero time later, it would be at point B. The time between it being at A and B would depend on how far apart they are.

It depends on the frequency of alternation. If the observer is A, then his clock is sure to be sacrilegious, but operational. He is as dead alive at a time.
In addition you can insert several rounds of altermation before taking a first step for the time.

1 hour ago, Strange said:

3. The mass would never be at different places at the same time, so I don't think the phrase "distribute the mass in several separate places" really means anything. The mass is no more "distributed in several separate places" than my mass is when I drive to work and back.

(That is the practical aspect. Can't help with the math, as I don't understand what you are trying to do.)

kartazion-hologramme.png

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1 hour ago, Kartazion said:

The velocity of quarks within the nucleus is nearly equal to that of light.

The key word there is "nearly".

Neutrinos also move at nearly the speed of light.

They do not move at the speed of light. Nothing with mass can.

 

 

17 minutes ago, Kartazion said:

It depends on the frequency of alternation. If the observer is A, then his clock is sure to be sacrilegious, but operational. He is as dead alive at a time.
In addition you can insert several rounds of altermation before taking a first step for the time.

Did you use google translate for that? I'm afraid the meaning didn't come across at all.

If an object moves at (or nearly at) the speed of light, we still see it moving between A and B (at the speed of light). So it will never be seen as two different objects. It will be observed as one object moving between A and B at the speed of light.

19 minutes ago, Kartazion said:

kartazion-hologramme.png

You need to explain what that diagram means. None of the words make any sense by themselves.

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8 minutes ago, Strange said:

The key word there is "nearly".

Neutrinos also move at. the speed of light.

They do not move at the speed of light. Nothing with mass can.

Yes I know. But in math we can go faster than c. But we can stick to it nearly. It looks fine to me.

Edited by Kartazion
spelling mistake

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Just now, Kartazion said:

Yes I know. But in math we can go faster than c. But we can stick to it nearly. It looks fine to me.

Still doesn't change everything else that is wrong in your claims.

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4 minutes ago, Strange said:

Still doesn't change everything else that is wrong in your claims.

I wish to mathematically exploit the alternation of a particle between two points at 1000 times the speed of light.

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1 minute ago, Kartazion said:

I wish to mathematically exploit the alternation of a particle between two points at 1000 times the speed of light.

Unless it is infinitely fast, it will still appear to be a particle moving between two points, and not two particles present at each point.

And any such mathematics will have no basis in, or relation to, reality. So I'm not sure what the point is.

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8 minutes ago, Strange said:

Unless it is infinitely fast, it will still appear to be a particle moving between two points, and not two particles present at each point.

I do not understand. A particle that alternates from A to B at 1000 times the speed of light would be seen by our eyes as a single point?

 

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Just now, Kartazion said:

I do not understand. A particle that alternates from A to B at 1000 times the speed of light would be seen by our eyes as a single point?

 

Why?

It would just be seen as a particle that moves very fast.

("with our eyes"? We can't see single points, even if they are stationary. I am assuming you are talking about making measurements with appropriate tools.)

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2 hours ago, Kartazion said:

Yes, I wish in the long term (much later) to explode quantum states of type supperposition, or entanglement.
I even suspect that the provence of quantum chromodynamics has originated in the oscillation of a single moving particle.

!

Moderator Note

This thread has gone from asking a question to proposing unsupported claims, so I need to move it to the appropriate section. Please note the special rules for Speculations, and support your assertions rigorously. 

 

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