Dimensions vs. degrees of freedom

[Hornbein]

TL;DR Summary

Are dimensions the same thing as degrees of freedom?

Are dimensions the same thing as degrees of freedom?

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

 

[MevsEinstein]

Yes the second question is true if you are considering just the circumference. Now dimensions aren't the same thing as degrees of freedom, but they kind of have the same analogy as shapes and dimensions.

 

[jedishrfu]

Heres the wiki on degrees of freedom:

https://en.wikipedia.org/wiki/Degrees_of_freedom

In a 3D mechanics context, objects can have 6 degrees of freedom. They aren’t the same thing.

 

[PeroK]

Summary: Are dimensions the same thing as degrees of freedom?

No. A simple pendulum moves in 2D but has only one degree of freedom.

 

[Jarvis323]

In a 3D mechanics context, objects can have 6 degrees of freedom. They aren’t the same thing.

The configuration space is 3 dimensional, but the phase space is 6 dimensional.

No. A simple pendulum moves in 2D but has only one degree of freedom.

The configuration space is 1D, but the phase space is 2 dimensional, thus the dynamical system has 2 degrees of freedom.

 

[PeroK]

The configuration space is 1D, but the phase space is 2 dimensional, thus the dynamical system has 2 degrees of freedom.

The second degree of freedom must be in your imagination!

 

[hutchphd]

The second degree of freedom must be in your imagination!

This issue is unclear in my head. To specify the "state vector" for a simple point 1D pendulum requires two numbers . Both the position and momentum appear quadratically in the Hamiltonian so equipartition applies to both.
For a point free particle only the velocity appears for equipartition. So what is the actual specification of DOF?

 

[Jarvis323]

The second degree of freedom must be in your imagination!

The second degree of freedom is momentum.

 

[PeroK]

The second degree of freedom is momentum.

Are you talking about a damped pendulum?

 

[PeroK]

The second degree of freedom is momentum.

You're saying that a particle moving with (undamped) SHM in 1D and a particle moving chaotically in 1D both have two degrees of freedom?

In other words, you're saying that the constraint in terms of the relationship between position and momentum in SHM does not result in a reduction in the number of degrees of freedom?

 

[PeroK]

This issue is unclear in my head. To specify the "state vector" for a simple point 1D pendulum requires two numbers . Both the position and momentum appear quadratically in the Hamiltonian so equipartition applies to both.
For a point free particle only the velocity appears for equipartition. So what is the actual specification of DOF?

I can't find any corroboration that a simple pendulum has two degrees of freedom. A double pendulum has two degrees of freedom according to, for example:

http://www.maths.surrey.ac.uk/explore/michaelspages/Double.htm

 

[malawi_glenn]

What if take into account the color of the pendulum? It has 3 dof then?

 

[Jarvis323]

You're saying that a particle moving with (undamped) SHM in 1D and a particle moving chaotically in 1D both have two degrees of freedom?

In other words, you're saying that the constraint in terms of the relationship between position and momentum in SHM does not result in a reduction in the number of degrees of freedom?

Each of your phase space variables are degrees of freedom. I guess parameters are too. So the number of degrees of freedom would be the number of dimensions of the phase space (for Hamiltonian system that is configuration space/generalized coordinates and velocity or momentum), plus the number of dimensions of the parameter space.

I believe you need at least a 3D phase space for a continuous system to be chaotic.

 

[Jarvis323]

I can't find any corroboration that a simple pendulum has two degrees of freedom. A double pendulum has two degrees of freedom according to, for example:

http://www.maths.surrey.ac.uk/explore/michaelspages/Double.htm

This source is also talking about the number of dimensions of the configuration space as a number of degrees of freedom, which is misleading I think, but I guess it can be true if you are talking about the number of degrees of freedom in just configuration space instead of the total number of degrees of freedom of the system.

 

[PeroK]

Each of your phase space variables are degrees of freedom. I guess parameters are too. So the number of degrees of freedom would be the number of dimensions of the phase space (for Hamiltonian system that is configuration space/generalized coordinates and velocity or momentum), plus the number of dimensions of the parameter space.

Do you have a reference where the simple pendulum is described as having two degrees of freedom. Everything I can find online says one degree of freedom.

 

[PeroK]

I believe you need at least a 3D phase space for a continuous system to be chaotic.

What's the third degree of freedom?

 

[Jarvis323]

Do you have a reference where the simple pendulum is described as having two degrees of freedom. Everything I can find online says one degree of freedom.

Read my previous post. I suspect you're using the term degrees of freedom in a limiting context (configuration space) which doesn't capture the full phase space. As an example, the equations for the double pendulum that you linked has a 4D phase space. So I would say that system has 4 degrees of freedom. Well, more actually since you should include the parameters according to the definitions I've seen.

 

[Jarvis323]

What's the third degree of freedom?

You said something about a chaotic 1d system, and I was just pointing out that a continuous system cannot be chaotic unless it has at least a 3 dimensional phase space. Not to say a system with only a 1D configuration space can't be chaotic.

 

[Jarvis323]

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

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

I would say, in terms of the degrees of freedom, for the equation of a circle, it should capture all of the information one would need to draw it. That would be the center and the radius, so 3 degrees of freedom total, or more if you consider color.

But also the space of circles has 3 dimensions. Any time there is a degree of freedom, I think that degree of freedom can be described as an ability to vary in a dimension.

So I think there is a subtleness about the differences between the terms. For example, if a space is static, like Euclidean space, I wouldn't say that space has degrees of freedom, but it does obviously have dimensions. But if the space can be distorted or varied, or there is a family of spaces based on some parameters, then you could talk about the degrees of freedom.

Likewise, it might be more precise to say a system can vary in N dimensions or is N dimensional rather than to say it has N dimensions. But I don't think enough people care about that level of specificity in language as long as they can understand each other well enough.

That's my understanding at least.

 

[PeroK]

I would say that a circle has 360 degrees of freedom!

 

[jedishrfu]

and as an aside Kevin Bacon has 6 degrees of separation:

https://blogs.ams.org/mathgradblog/2013/11/22/degrees-kevin-bacon/

 

[jbriggs444]

But also the space of circles has 3 dimensions. Any time there is a degree of freedom, I think that degree of freedom can be described as an ability to vary in a dimension.

That is the space of circles in a plane. If you want circles in 3-space, you need one more degree of freedom for the location of the center and two more degrees of freedom to nail down the orientation of the plane.

But if you want to track the location of a bug on that circle, that's only one degree of freedom. Two if you want its velocity as well.

 

[robphy]

Heres the wiki on degrees of freedom:

https://en.wikipedia.org/wiki/Degrees_of_freedom

In a 3D mechanics context, objects can have 6 degrees of freedom. They aren’t the same thing.

In my opinion, it is clear that
on their own
"dimensions" and "degrees of freedom"
are not well defined...
but instead are convenient shortcuts in some context.

In the wikipedia article linked above

• Mechanics​

  ... A free, rigid object, such as a ship at sea, has six degrees of freedom: three rotations and three translations about each perpendicular axis.

  describes the dimensionality of the configuration space

• Physics and chemistry​

  ...or the dimension of its phase space, is known as its degrees of freedom

  describes the dimensionality of the phase space

So, it appears there is no official unambiguous definition.
Thus, it's probably best to clearly state
"Here, we define the number of degrees of freedom to be dimensionality of the _____."

(Note that constraints can decrease the dimensionality:
e.g. a point-particle constrained to travel on a circle
has a smaller dimensionality than an unconstrained one.)