Dimensions Vs. Ranges Of Freedom

IThread starterHornbeinStart dateJul sixteen, 2022SummaryAre dimensions the equal thing as tiers of freedom?

Are dimensions the identical component as tiers of freedom?

Would you are saying that a circle is a one dimensional object embedded in adimensional space?Answers and Replies

Yes the second query is true if you are thinking about simply the circumference. Now dimensions are not the same factor as levels of freedom, however they sort of have the identical analogy as shapes and dimensions.

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Summary: Are dimensions the identical thing as ranges of freedom?

No. A simple pendulum moves in 2D however has best one diploma of freedom.

In a three-D mechanics context, gadgets will have 6 stages of freedom. They aren’t the equal issue.

The configuration area is three dimensional, however the segment area is 6 dimensional.

No. A simple pendulum movements in 2D however has handiest one diploma of freedom.

The configuration area is 1D, however the section space is two dimensional, hence the dynamical machine has 2 levels of freedom.

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The configuration space is 1D, however the segment area is two dimensional, accordingly the dynamical device has 2 stages of freedom.

The 2nd diploma of freedom ought to be in your creativeness!

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The 2nd degree of freedom have to be for your imagination!

This issue is unclear in my head. To specify the “kingdom vector” for a easy factor 1D pendulum requiresnumbers . Both the position and momentum appear quadratically inside the Hamiltonian so equipartition applies to both. For a point unfastened particle best the velocity seems for equipartition. So what is the real specification of DOF?

The 2d diploma of freedom ought to be in your imagination!

The second diploma of freedom is momentum.

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The 2nd degree of freedom is momentum.

Are you speaking approximately a damped pendulum?

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The 2nd degree of freedom is momentum.

You’re saying that a particle shifting with (undamped) SHM in 1D and a particle moving chaotically in 1D each have two stages of freedom?

In other phrases, you’re pronouncing that the constraint in phrases of the relationship among role and momentum in SHM does no longer result in a reduction within the number of tiers of freedom?

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This trouble is doubtful in my head. To specify the “state vector” for a easy factor 1D pendulum calls fornumbers . Both the location and momentum seem quadratically within the Hamiltonian so equipartition applies to both. For a point loose particle handiest the rate seems for equipartition. So what is the real specification of DOF?

I can not find any corroboration that a simple pendulum has two degrees of freedom. A double pendulum has two levels of freedom consistent with, for instance:

http://www.maths.surrey.ac.united kingdom/discover/michaelspages/Double.htm

What if don’t forget the colour of the pendulum? It has 3 dof then?

You’re announcing that a particle moving with (undamped) SHM in 1D and a particle shifting chaotically in 1D both have two ranges of freedom?

In other phrases, you’re announcing that the constraint in terms of the connection among role and momentum in SHM does now not result in a reduction within the quantity of tiers of freedom?

Each of your phase area variables are tiers of freedom. I bet parameters are too. So the variety of degrees of freedom will be the quantity of dimensions of the phase area (for Hamiltonian system this is configuration area/generalized coordinates and speed or momentum), plus the wide variety of dimensions of the parameter area.

I accept as true with you need as a minimum a 3-D phase space for a non-stop machine to be chaotic.

I can not find any corroboration that a easy pendulum hastiers of freedom. A double pendulum hastiers of freedom in step with, for example:

http://www.maths.surrey.ac.united kingdom/discover/michaelspages/Double.htm

This source is likewise speakme approximately the wide variety of dimensions of the configuration space as some of stages of freedom, that is deceptive I assume, however I bet it can be proper if you are talking approximately the variety of levels of freedom in just configuration area in place of the full variety of stages of freedom of the device.

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Each of your segment area variables are tiers of freedom. I bet parameters are too. So the wide variety of degrees of freedom would be the wide variety of dimensions of the section area (for Hamiltonian system this is configuration area/generalized coordinates and pace or momentum), plus the variety of dimensions of the parameter area.

Do you have a dobrovol.org reference wherein the easy pendulum is defined as havingranges of freedom. Everything I can locate on line says one diploma of freedom.

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I accept as true with you want as a minimum a 3-D phase area for a continuous device to be chaotic.

What’s the third diploma of freedom?

Do you have a reference in which the simple pendulum is described as having two degrees of freedom. Everything I can find on line says one diploma of freedom.

Read my previous put up. I suspect you are the use of the term ranges of freedom in a limiting context (configuration area) which does not seize the overall segment space. As an instance, the equations for the double pendulum that you related has a 4D segment space. So I might say that system has four stages of freedom. Well, more certainly since you must consist of the parameters consistent with the definitions I’ve visible.

Last edited: Jul 17, 2022

What’s the 1/3 diploma of freedom?

You said something approximately a chaotic 1d gadget, and I become just pointing out that a continuous gadget cannot be chaotic unless it has as a minimum a 3 dimensional phase area. Not to say a gadget with most effective a 1D configuration space cannot be chaotic.

Last edited: Jul 17, 2022

Would you say that a circle is a one dimensional object embedded in a two dimensional area?

A circle is a 1D manifold embedded in a 2D space.

I would say, in phrases of the stages of freedom, for the equation of a circle, it need to seize all of the records one could need to draw it. That will be the middle and the radius, so three ranges of freedom general, or more if you do not forget coloration.

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