Degrees of freedom and falling chalk

[Rasalhague]

In lecture 2 of his series on Classical Physics, Prof. Balakrishnan asks how many degrees of freedom there are in a system consisting of a falling piece of chalk. Looking at my notes, I'm not sure I understand the concept. Is the answer 6 (3 of position, 3 of orientation), or 12 (3 of position, three of orientation, 3 of linear velocity, 3 of angular velocity), or something else? Is the number of degrees of freedom the dimension of the configuration space, or of the phase space? Is there always a natural distinction between configuration space and phase space in dynamical systems?

 

[gsal]

before things get blown out of proportion I would consider what the constraints of the system are and the assumptions etc

for example, if I don't really care about the orientation of the piece of chalk...those go away

if I know from which height the chalk started to fall and I know that my system ONLY consists of the chalk, the ground and gravity and nothing else is allowed into the system (external objects, forces, etc) ...

...then, I think my system has 1 degree of freedom, TIME

tell me that time and I will tell you at what height the piece of chalk is and how fast is going (or whether already hit the ground)...

my 2 cents

 

[Rasalhague]

if I know from which height the chalk started to fall and I know that my system ONLY consists of the chalk, the ground and gravity and nothing else is allowed into the system (external objects, forces, etc) ...

...then, I think my system has 1 degree of freedom, TIME

tell me that time and I will tell you at what height the piece of chalk is and how fast is going (or whether already hit the ground)...

I think your point is that the number of degrees of freedom depends on how much detail we want to include in our model? In your system, there's one degree of freedom. I guess we could measure it by time or by position, since either determines the other. But what if we have a system where the chalk is treated as a rigid body, having at least 6 degrees of configurational freedom, is it the convention for components of angular and linear velocity to count among degrees of freedom, or does the term degrees of freedom only refer to the number of possible configurations?

 

[Studiot]

Good evening, resalhague.

Surely you need to specifiy the discipline the degrees of freedom are involved in?

For example chalk is a simple molecule and if it is falling under gravity through a solution of acid there is only one site (degree of freedom) at which the acid can attack the molecule.

 

[gsal]

then I think everyone counts, I mean, even every component of a given quantity like position or velocity.

It's been a while since I have properly addressed a problem (I just solve them ) but if I recall correctly, for me, a degree of freedom is basically a state variable and the set of state variables is the independent set of quantities that fully determine the state of the system...so, if you want to take into account the position of your object in the 3D space and its orientation, then, yes, everyone of those dimensions count.

 

[jambaugh]

I think the best answer is six. There is some contextual ambiguity but generally when we enumerate degrees of freedom, say to determine the thermodynamic properties of a gas based on atomic degrees of freedom, we count the configuration space dimension (minus constraints if any).

Knowing the configuration dimension = degrees of freedom tells us the phase-space dimension so counting twice is redundant. But in an other context, say mathematics, where one is counting free parameters to solve a system of equations then the phrase may be used differently and have a different answer.

Here's an example of Classical Mechanics lecture notes which counts degrees of freedom:http://www.scribd.com/doc/3379962/Lectures-on-Classical-Mechanics1"

 

[Rasalhague]

Thanks, all. And thanks, jambaugh, for answering my question : ) Nice link.

 

[niyazkc88]

CAn you tell me how it can be 6 degrees?
There are 3 degrees of rotational freedom and one degree of chalk going up or down.
So that would be 4, right?
Please correct my ignorance if I am wrong..

 

[Studiot]

Welcome to Physics Forums

Freedom says it all

There is nothing in the system to prevent the chalk moving in either of the available sideways directions.

go well

 

[Rasalhague]

When I said 6, I was assuming there were no contraints placed on initial velocity. If it could be moving in any direction, with any speed at the outset, then it's free to move in a 3d configuration space. But if the initial velocity was constrained to be zero, then I'm guessing it would have 4 degrees of freedom, as you said. (But I'm no expert, so I could be mistaken.)

According to Balakrishnan, a holonomic constraint means one which takes away a degree of freedom, whereas Gupta's notes (jambaugh's link) say, "A Holonomic constraint is one that can be expressed in the form of an equation relating the coordinates, $F(x_1,y_1,z_1,...,t)=0$". I think, by coordinates, Gupta means the component functions of a vector-valued function of real numbers whose range is a possible trajectory through configuration space: $x_1:I\subseteq \mathbb{R}\rightarrow \mathbb{R}$ etc. with $t \mapsto (x_1(t),y_1(t),z_1(t),x_2(t),...,Id(t)=t)$. I base this conclusion on the fact that he differentiates them with respect to time and writes the derivatives with straight, rather than curly "d". Constraints, it seems, can include, as in his final example, conditions placed on the derivatives of these component functions.

By the way, is it possible to see more than the first two pages of Gupta's "Lectures on Classical Mechanics"? I'm using Firefox, which cuts the text off abruptly at the end of page 2, just as he's about to introduce an example of holonomic and scleronomic constraints.

 

[Studiot]

When I said 6, I was assuming there were no contraints placed on initial velocity. If it could be moving in any direction, with any speed at the outset, then it's free to move in a 3d configuration space. But if the initial velocity was constrained to be zero, then I'm guessing it would have 4 degrees of freedom, as you said.

What do you mean constrained to be zero?

This is not really the meaning of constraint. A constraint would stay in effect, regardless of the velocity or positiion of the falling chalk.

The basic number of degrees of freedom of mechanical motion available are 6.

Each true constraint removes one of them.

Some constaints might be the chalk being like a bead sliding down a wire so it cannot move sideways, or rotate about certain axes.

eg

Chalk falling freely

movement in x direction yes
movement in y direction yes
movement in z direction yes
rotation about x direction yes
rotation about y direction yes
rotation about z direction yes

Total degrees of freedom 6
No of constaints 0

Chalk sliding down a wire

movement in x direction no
movement in y direction no
movement in z direction yes
rotation about x direction no
rotation about y direction no
rotation about z direction yes

Total degrees of freedom 2
No of constraints 4

go well

 

[niyazkc88]

Mr. Studiot, Isnt a chalk falling freely under gravity constrained by it to fall in only one direction? Falling through a wire vertically still one direction, but the constraint is still not imposed by wire but by gravity.
Gupta's notes say."The no. of degrees of fr is n=3N-K, where N=no of particles and k is no of constraint equations."
Can you tell me Mr. RASALHAGUE, how many constraint equations are there for the chalk? Would it be y=0 and x=0?
And can the chalk be seen to be made of 2 particles?

The third page of Gupta's notes include examples on a disc rolling on the xy plane and a sphere rolling on a larger sphere. Can you guess the types of constraints on both these examples?
I have not understood the notes correctly, I have 2 read them again and again..

 

[Studiot]

Hello, niyazkc88.

No you have not appreciated what a constraint is.

Nor have you noted my post#9.

A degree of freedom is just that. It is a statement that the system is able to move in the manner prescribed by that particular degree.

It does not say that the system will move in that manner, just that it could do so if suitable external forces (loads ) were applied.

A constraint is the opposite it says that, even if external forces or loads were applied the system still could not and would not move in that particular manner.

So with the piece of chalk example, if there were a side wind blowing the chalk would also move sideways as it fell freely.

But the chalk strung like a bead on a wire could not.

It is very important to realize the force causing the motion is not the constraint.

Hope this clears it up for you.

 

[niyazkc88]

Yes, I think you are right. Thanks for clearing my mind.
A planet moving around the sun is not be constrained to move in a circle or ellipse. Is that right?

 

[Studiot]

Yes we do also say 'constrained to move in an ellipse', but that is really loose wording and can lead to the confusion with the use of 'constraint' in this particular specific meaning.

You question about two lumps is a good one, however.

The degrees of freedom also depend upon the shape and configuration of the system.

All bodies have just the 6 motional degrees maximum.

However if we consider vibrational modes as well that introduces many more.

The piece of chalk has only one lump and one vibrational mode - expanding and contracting like a pulsating sphere.

A system of two or more lumps joined by a rods and/or springs has flapping, twisting, stretching modes.

go well

 

[Rasalhague]

What do you mean constrained to be zero?

This is not really the meaning of constraint. A constraint would stay in effect, regardless of the velocity or positiion of the falling chalk.

The basic number of degrees of freedom of mechanical motion available are 6.

Each true constraint removes one of them.

jambaugh has told us that the number of degrees of freedom is the dimension of the configuration space. According to Balakrishnan's definition, a holonomic constraint is a condition that removes a degree of freedom. According to Gupta's, a holonomic constraint is a condition that can be expressed as an equation relating "coordinates", by which he appears to mean the component functions of a vector-valued function whose range is a trajectory through the configuration space.

By velocity constrained to be zero, I mean that a condition is imposed requiring that the derivatives of all of the functions whose range is a trajectory through the configuration space associate the output value 0 with the input t = 0. If the system would otherwise have had 6 degrees of freedom - 3 of position, 3 of orientation - then it seems like the imposition of this condition would reduce the degrees of freedom to 4, as niyazkc88 said. So the condition meets Balakrishnan's definition of holonomic constraints. It also meets Gupta's, since we can express them as dx/dt = dy/dt = dz/dt = 0. There are two equations here which relate a pair of "coordinates", and - sure enough - they remove two degrees of freedom.

But maybe you're saying the convention is to not call this a constraint? If so, I really don't understand yet.

 

[Rasalhague]

Mr. Studiot, Isnt a chalk falling freely under gravity constrained by it to fall in only one direction? Falling through a wire vertically still one direction, but the constraint is still not imposed by wire but by gravity.
Gupta's notes say."The no. of degrees of fr is n=3N-K, where N=no of particles and k is no of constraint equations."
Can you tell me Mr. RASALHAGUE, how many constraint equations are there for the chalk? Would it be y=0 and x=0?
And can the chalk be seen to be made of 2 particles?

The third page of Gupta's notes include examples on a disc rolling on the xy plane and a sphere rolling on a larger sphere. Can you guess the types of constraints on both these examples?
I have not understood the notes correctly, I have 2 read them again and again..

I'm considering the "chalk" to be a rigid body, meaning that the distance between every pair of its constituent particles is fixed. That's just the question I'm considering, for the sake of an example, to try to clarify the meaning of the expression "degrees of freedom". Of course, we could consider different examples, in which "chalk" means something different. In retrospect, it would have been clearer if I'd said rigid body at the outset.

If we start off considering a configuration space which is the set of all positions and orientations of a rigid body, there are 6 degrees of freedom. If we end up, somehow, with 4 degrees of freedom, I guess - from what Gupta writes - that we'd would need two equations relating "coordinates". Imposing the condition that the only possible initial velocity is zero would imply that dx/dt = dy/dt, and that dy/dt = dz/dt. That's two equations. I'm just trying to learn this material myself though, so I could be mistaken.

 

[Studiot]

I think a basic difficulty is the use of the terms

'degree of freedom'
and
'constraint'

The former means pretty much the same whatever discipline you are discussing, although the actual freedom may vary, as I demonstrated with my first reply.

The word constraint has two different distinct meanings or uses.

1) As a counterpart blocking or removing a degree of freedom.

2) In the sense of a boundary or intitial condition in a (often differential) equation or set of equations.

Both uses are very common and need to be distinguished by the context.

So the statement

'The initial velocity is zero or constrained to zero' is an example of meaning 2 and not a statement about the degree of freedom enjoyed by the object under consideration.
It is implied in this statement that although v may be initially zero, it may very well be nonzero at some other time or place. It obviously cannot be so without the freedom to do so.

However

'The velocity is constrained to zero' is a statement about the velocity at any and all times and places and is indeed a restriction in the degrees of freedom (3 in fact since velocity has 3 components)

Of course you can choose other variables to make the statements about, but all possibilities are fully covered by the minimum set, you don't gain additional degrees just by considering additional variables.

go well

 

[Rasalhague]

I think a basic difficulty is the use of the terms

'degree of freedom'
and
'constraint'

The former means pretty much the same whatever discipline you are discussing, although the actual freedom may vary, as I demonstrated with my first reply.

Except that, if I understood jambaugh, in classical mechanics, degrees of freedom is the dimension only of the configuration space, whereas in other contexts it's simply the dimension of the whole state space?

The word constraint has two different distinct meanings or uses.

1) As a counterpart blocking or removing a degree of freedom.

2) In the sense of a boundary or intitial condition in a (often differential) equation or set of equations.

Both uses are very common and need to be distinguished by the context.

So the statement

'The initial velocity is zero or constrained to zero' is an example of meaning 2 and not a statement about the degree of freedom enjoyed by the object under consideration.
It is implied in this statement that although v may be initially zero, it may very well be nonzero at some other time or place. It obviously cannot be so without the freedom to do so.

However

'The velocity is constrained to zero' is a statement about the velocity at any and all times and places and is indeed a restriction in the degrees of freedom (3 in fact since velocity has 3 components).

Thanks for your explanation. Is your "constraint sense 2" what is meant by nonholonomic constraints? Balakrishnan said a nonholonomic constraint is one which doesn't reduce degrees of freedom. He gave, as an example, the restriction of "coordinates" to some subset of the real numbers. Would "for all t, x'(t) = 0" be a holonomic constraint? It's an equation involving x, which meets your condition of holding for all values in the domain of x, but it doesn't relate x to another "coordinate", as Gupta seems to require. Would "for all t, x'(t) = y'(t) be a holonomic constraint?

 

[Studiot]

Except that, if I understood jambaugh, in classical mechanics, degrees of freedom is the dimension only of the configuration space, whereas in other contexts it's simply the dimension of the whole state space?

First, let me apologise for going off thread in answering niyazkc88, his question was about a much simpler view of mechanics.

Now I realize your question is about generalised coordinates I can comment.

Yes the degree of freedom is the dimension of the configuration space in classical mechanics.

Further the DOF + number of constraints = dimension of the state space, agreed.

I don't think Jambaugh was lumping the DOF and constraints together and calling them some alternative version of DOF, but I'm sure he can speak for himself.

However, more fundamentally than a bit of semantics is what I meant by the difference between 'constraints' as strictly defined here and equations, as exemplified by your professor's chalk example.

The chalk could have an initial velocity of zero or it could be given an initial push (or even a continued one). These situations would lead to different values of position against time(different numerical solutions to the equations of motion), but would not change the DOF of the chalk.

Does this make sense?

 

[Rasalhague]

Further the DOF + number of constraints = dimension of the state space, agreed.

I don't think Jambaugh was lumping the DOF and constraints together and calling them some alternative version of DOF, but I'm sure he can speak for himself.

However, more fundamentally than a bit of semantics is what I meant by the difference between 'constraints' as strictly defined here and equations, as exemplified by your professor's chalk example.

[...]

Does this make sense?

Not entirely.

Did you have any thoughts on the specific questions I asked at the end of my last post? You say there are multiples uses of the word constraint. Here it must be your constraint₁, which I think is what other people call a holonomic constraint, but I don't know whether you agree with this identification, or disagree, or are unsure.

I think we're talking at cross purposes with respect to state space. I'm using state space very generally to mean the set of all possible states of a system being considered. In a dynamical system, I presume this includes the values of generalised velocity as well as configurations. Balakrishnan calls it phase space, but I've read that phase space refers specifically to the case where the dynamical system has independent variables called position and momentum. In his examples so far, the independent variables (=coordinates of arbitrary points) have been called position and momentum. But maybe that's a trivial difference. Maybe I should call it phase space. But state space seem to have greater generality, being applied to other problems besides classical mechanics.

I think you're using state space in the following sense. Suppose we have one configuration space. Suppose we decide this model contains a lot of redundancy/superfluity when we apply it to a given physical system. Suppose we want to get rid of that redundancy by imposing the sort of constraints that reduce degrees of freedom (perhaps this is equivalent to: imposing conditions specified by a set of equations relating dynamical variables (=coordinates of paramaterizations that describe trajectories in Rasalhague state space = Balakrishnan's phase space). Then the original configuration space is Studiot state space. Or perhaps it refers specifically to the unique, hypothetical original which accounts for every single particle in the system? I think the semantics (meaning!) is important. If we ignore it, we'll get confused, and confuse each other : )

I agree that jambaugh didn't mean what you say he didn't mean! I never imagined he did. He wrote: "But in an other context, say mathematics, where one is counting free parameters to solve a system of equations then the phrase may be used differently and have a different answer."

 

[Studiot]

Is x'(t)= 0 $\forall$t Holonomic or not?

Well the definition of holonomic is that you can make arbitrary changes to the variables without constraint.

I would say the $\forall$t is a constraint, wouldn't you?

 

[jambaugh]

I don't think Jambaugh was lumping the DOF and constraints together and calling them some alternative version of DOF, but I'm sure he can speak for himself.

Yes, in a way I was. It depends on how we begin setting up a given system and our aim.

Example: If we are considering a rigid pendulum say a bar hanging from a frictionless ball socket, then we can either count the degrees of freedom for a free bar then consider the constraint imposed by the mechanical connection separately, or we can cut to the chase and consider only the two DOF of the pendulum, e.g. the two angular components.

It is all in the wording of the system definition, is it "a bar pinned at a point" or is it a "pendulum". The mechanics is the same but definitions of terms may vary slightly in context as one considers which more general systems the one under study is to be incorporated into. Say if one wants to consider relaxing a constraint or finding a force of constraint then one may want to start with the more general base system and include the extra degrees of freedom.

I mean ultimately a rigid rotating body is a collection of atoms with 3N degrees of freedom and 3N-6 constraints. It is silly to start there if you want to treat it as a RR... but if you want to generalize to a rotating elastic body you find new DOF's from amoung those 3N, enumerate modes you wish to consider and ignore the rest.