Degree of Freedom: Definition & Examples - Confused? Ask Here!

[LCSphysicist]

Homework Statement

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Relevant Equations

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Hello. I am a little confused with the definition of degree of freedom, since i always count it wrong when it is necessary:
Is it the number of coord?inates necessary to describe the problem, or the number of independent coordinates

I ask this because, for example, see this problem:
A rotator with a symmetric axis + a particle. Both interacting via a potential V(|r-R|). How many degree of freedom the system have?
The answer, second the book, is 9.
BUT, i would say that the center of mass is fixed, and since the rotator is symmetric, rotation about its axis is useless. So we have $$9 - 1 (CM) - 1 = 7$$. 7 coordinates necessary to describe the motion, so 7 Degree of freedom? Namely, $$x,y,z,X,Y,\theta,\phi$$ (Z is determined by CM position).

What am i interpretating wrong?

 

[Orodruin]

Why would the center of mass be fixed? (If it were then you would have to remove three spatial coordinates.)

That an object has a symmetric axis does not mean it cannot rotate about that axis. Consider a disk spinning around its center of mass around an axis perpendicular to it. By your reasoning that should have zero degrees of freedom but it certainly does not.

 

[LCSphysicist]

Why would the center of mass be fixed? (If it were then you would have to remove three spatial coordinates.)

That an object has a symmetric axis does not mean it cannot rotate about that axis. Consider a disk spinning around its center of mass around an axis perpendicular to it. By your reasoning that should have zero degrees of freedom but it certainly does not.

Oops :( I assumed that ausence of external force implies the center of mass does not move. DAmn Aristoteles XD Sometimes this fools me.

Ok, so the degree of freedom in general are reduced only by geometrical constraints?

 

[hutchphd]

Homework Statement:: .
Relevant Equations:: .

since the rotator is symmetric, rotation about its axis is useless.

You need to be careful here also. The statement is that the rotor has an axis of symmetry not that it has circular symmetry (about that axis). The complete circular symmetry would obviate that degree of freedom.

 

[Lnewqban]

Is it the number of coordinates necessary to describe the problem, or the number of independent coordinates?

Please, see:
https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

"The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify a configuration."

 

[Orodruin]

The complete circular symmetry obviate take away that degree of freedom.

You are arguing that the disk described above in #2 has zero degrees of freedom?

 

[hutchphd]

I think so. How could it couple ?

 

[Steve4Physics]

Hello. I am a little confused with the definition of degree of freedom ...
Is it the number of coord?inates necessary to describe the problem, or the number of independent coordinates

The number of DoFs is the minimum number of coordinates needed to describe the configuration of a system. The coordinates must be independent.

since the rotator is symmetric, rotation about its axis is useless

The purpose of a rotator is to rotate. Or you would not deliberately specify a rotator! Consequently, the angle of rotation is an essential parameter (as opposed to ‘useless’!).

Maybe pretend that the (symmetrical) rotator has a dot of paint somewhere off-axis. (And do rotators have to be symmetric?)

Assuming no constraints:
How many parameters are needed to determine the position of the centre of mass of the rotator?
How many parameters are needed to determine the orientation of the rotator?
How many parameters are needed to determine the position of the particle?
What’s the total?

EDIT. Aha, didn't see the recent posts while composing the above.

 

[Delta2]

What I never understood about degrees of freedom is how we can have systems with more than 3 degrees of freedom since the maximum number of coordinates is 3 (at least in classical physics we are in 3D space +1 dimension for time).

Ok if I understand it now, the coordinates we talk about are not the coordinates of the system of reference but something like the generalized coordinates in Lagrangian Mechanics.

 

[Steve4Physics]

What I never understood about degrees of freedom is how we can have systems with more than 3 degrees of freedom since the maximum number of coordinates is 3 (at least in classical physics we are in 3D space +1 dimension for time).

Ok if I understand it now, the coordinates we talk about are not the coordinates of the system of reference but something like the generalized coordinates in Lagrangian Mechanics.

Yes. We (as physics students) probably first met the term 'degree of freedom' when learning about ideal gases. E.g. for an ideal gas of each (point) particle has 3 degrees of freedom.

But the term 'degree of freedom' is much more widely used. For example in robotics, the following machine has 5 degrees of freedom. Some parameters (e.g. arm-lengths) are fixed but some parameters (5 angles) are not:
https://www.researchgate.net/publication/319127421/figure/fig2/AS:631651373695060@1527608831974/Five-degrees-of-freedom-robot-arm-model.png

 

[Orodruin]

I think so. How could it couple ?

By this argument a disk rotating about its center will have no kinetic energy or angular momentum.

 

[hutchphd]

We regularly ignore most of the universe when we define an "isolated" system. The argument I would put forward is that these are not mutable.