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Newton's Laws: Internal Degrees of Freedom

  1. Apr 22, 2014 #1
    Hello Everyone,

    I am currently re-reading Taylor's Classical Mechanics, in particular, section 4 in chapter 1. He is discussing Newton's Laws, and makes the statement:

    "A point mass, or particle, is a convenient fiction, an object with mass, but no size, that can move through space but has no internal degrees of freedom."

    What are internal degrees of freedom, and how do they compare to "ordinary" degrees of freedom?
  2. jcsd
  3. Apr 22, 2014 #2
    I assume that means it is a rigid body that can't deform in any manner.
  4. Apr 23, 2014 #3


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    DOF=degrees of freedom. The number of variables required to describe its possible motions.

    For example, the point particle can only move in the three dimensions of space, so DOF=3.
    If we have two of them, each contributes 3, so DOF=6.
    Now tie them together with a "rod"; we can now track them via the center of mass, which has 3 DOF like any point, plus two angles for orientation - hence DOF=5. Or we can just note DOF=6 - 1 due to the constraint of the rod.

    If the rod is a spring, then DOF=6 because of vibrations.

    Now make the points into particles with size (atoms or rocks); this adds additional DOF because now they can also rotate about the axis which connects them; this adds an additional angle, hence one more DOF.

    A rigid body - such as an ideal rock - will thus have DOF=6.
  5. Apr 23, 2014 #4


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    The basic idea is that a point particle doesn't have any internal "structure", so there are no additional mathematical variables needed to describe what is going on inside it.

    Often the idea of a point particle is a good approximation for something "small" that does have an internal structure in real life. For example a small flexible object can have an internal stresses and strains, an internal temperature distribution, etc. But modeling it as a point particle excludes all those things from the model, and therefore you have to be careful not to ask questions about the physics that are sensible, but which the model can't answer. (The map is not the territory!)
  6. Apr 23, 2014 #5
    These would be external or "ordinary" degrees of freedom. The OP was asking about the meaning of internal degrees of freedom.
  7. Apr 23, 2014 #6


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    Like the book said, point particles "have no internal degrees of freedom" ... which is obvious if you (a) understand DOF, and (b) understand what is meant by a point particle.

    BTW, if the author of the OP has further questions, he is welcome to ask, as are you.
  8. Apr 23, 2014 #7
    It wasn't obvious to the OP since he asked about the meaning of internal degrees of freedom which I don't think you really addressed in your response.

    Also, a point mass actually has 6 degrees of freedom. Your post stated that it only has 3 degrees of freedom.
  9. Apr 24, 2014 #8


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    Really? I'm glad you know what the author of the OP is thinking; that is quite a talent.

    As for the DOF of a point particle - please show me how you would track all six of your proposed DOF for a point. I'll give you the first three: motion in the three dimensions of space.
  10. Apr 24, 2014 #9
    I didn't need to know what the OP was thinking, he explicitly asked it.

    If I weld a rod to the "point" then the rod can't rotate around that point unless the point has a rotational degree of freedom.
  11. Apr 24, 2014 #10

    Andrew Mason

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    First of all, you cannot weld anything to a point.

    Second, since a point has no spatial features, it is meaningless to say that a point can rotate. If you disagree, then define what you mean by a rotation of a point.

    Internal degrees of freedom are all of the degrees of freedom that a body has due to motion about its centre mass. The term does not include the translational degrees of freedom associated with the motion of its centre of mass.

  12. Apr 24, 2014 #11
    First of all, there is no such thing as a point mass. As the OP quoted above, it is a convenient fiction. So if you claim a point exists then I can also claim I can weld a fictitious line to it.

    Second, I will define rotation of a point to be the degree of freedom that allows a connected object to rotate around that point.

    In your definition, the centre of mass is a point. And this point has 6 degrees of freedom. So it is actually meaningful to say a point can rotate.
  13. Apr 24, 2014 #12

    Andrew Mason

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    You are actually defining the rotation of the connected object and just deeming the rotation of the point to be the same.

    I agree that a point mass is an idealized mass whose form does not have any spatial dimensions. I agree that it cannot really exist. No one is saying that a real body can be a point particle (except perhaps fundamental particles, such as electrons or quarks, in which case Newtonian mechanics does not apply).

    Since a point particle would have spatial dimensions of 0, its moment of inertia would be 0. Its angular momentum would be 0. Its rotational kinetic energy would be 0. There is no physical way to tell whether it has rotated. So I am not sure what you mean when you say it has rotated.

    This is not how degrees of freedom are defined.

    The centre of mass is a geometric point. It is not a point mass. The centre of mass does not rotate. It can't. It is just a point. A rotation means that the direction of a point in the body relative to the centre of the body has changed. The parts of the body rotate about its centre of mass. But the rotation of the centre point is undefined.

  14. Apr 24, 2014 #13
    No, I am defining the rotation of an object connected to an arbitrary point. And I claim I can tell whether the point has rotated if the object it was attached to rotated as well.

    I can idealize an object as a series of point masses connected rigidly (or not). Each one of those points have degrees of freedom associated with them. I can restrain the object from rotating by removing the rotational degrees of freedom of just one of those points including the center of mass. Therefore the points must have a rotational degree of freedom.
  15. Apr 24, 2014 #14
    your claim is still false. just b/c a rod or some other object rotates about an arbitrary point doesn't mean that the point rotates with it. it simply cannot be proven.

    take a circle in a 2-dimensional space and rotate it an arbitrary number of degrees/radians. the reason we "know" the circle has rotated X° or radians about its center is because we can track any one of an infinite number of points that make up the circle's circumference as it moves. try the same experiment with a sphere in 3 dimensions, and you'll see that its rotation can be tracked using any one of an infinite number of points that make up its surface.

    are you starting to see a pattern here? you need a point of reference on the border/surface of an object to track its rotation, which means you need to start with an object that has more than 0 dimensions. there is no point of reference on a point b/c its already a point and has no dimensions. therefore it has no rotational degrees of freedom. even if a point did "rotate," we have no way of knowing that it did b/c the rotation of a point cannot be measured. and so it is meaningless to describe the rotation of a point, since regardless of how many degrees/radians a point "rotates" through, its appearance/existence will always be the same to us.
  16. Apr 24, 2014 #15

    D H

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    A point mass only has three degrees of freedom, it's position in three dimensional space. The orientation of a point is undefined.

    You are dragging this off topic, paisello2. Stop.

    Back to the original post,
    Some examples: Internal vibrations in a solid, molecules of air colliding off one another. Those are both examples of internal degrees of freedom associated with thermodynamics.

    Just as a point mass does not have an orientation, it does not have a temperature.
  17. Apr 25, 2014 #16
    I would make a response to the above comments but now an administrator is involved in the argument. As he can arbitrarily delete posts and lock threads at his discretion, it is a waste of my time to make any responses in this thread.
  18. Apr 25, 2014 #17

    Andrew Mason

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    We can only say that a point has rotated relative to another point. If the displacement of point in a rigid body relative to the centre of mass of the body changes, then we can say that the body has rotated. We can also say that the point has rotated about the centre of mass. But we can't say that the point has rotated about itself. That has no meaning.

    Last edited: Apr 25, 2014
  19. Apr 25, 2014 #18
    We can argue about what a point mass is, but it's pretty clear what Taylor meant in the passage. A point mass has three degrees of freedom for the position in 3D space. (In phase space, it requires 6D to describe a point mass because you must also specify momentum.) The internal degrees of freedom are any moving parts inside the point mass if you zoom into it. Since there are no moving parts in the mass, there are no internal degrees of freedom. Internal means inside an object made up of parts which can be treated as a whole.

    The argument is merely over definition and nothing substantive. An electron is often called a point mass, but it does have an internal degree of freedom called spin. In a classical universe, you could imagine a rotating point as some kind of limit of a rotating object as the size went to zero, but we don't live in a classical universe.
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