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According to this Wikipedia article, the definition of degrees of freedom is:

The degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.

It seems to me that the degrees of freedom is mathematically ill-defined, unless I'm misunderstanding something. For instance,

  1. Can you count time as a parameter? If you know the equations of motion, the value of time would completely specify the system's configuration, so it would always have one degree of freedom.

  2. Can't you always increase the number of parameters (even if it has no effect) and still determine the configuration of the system?

  3. In the case of a pendulum, most texts say it has only one degree of freedom, but it can move any direction along the surface of a sphere (with radius equal to string length). So why shouldn't it have 2 degrees of freedom?

  4. In the case of projectile motion, the projectile has 3 degrees of freedom, right? But, can't we use the arc length along the path of motion as a coordinate, to fully specify its position? Doesn't this mean it has 1 degree of freedom?

Can someone give me a rigorous definition of the degrees of freedom and explain how this definition addresses the questions above?

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    $\begingroup$ Maybe something along the lines of: number of integration constants when you solve the equations of motion? Ordinarily, you get rid of the integration constants by applying some constraints that define the system's configuration at $t_0$. $\endgroup$
    – innisfree
    Jan 3 '16 at 23:57
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    $\begingroup$ I can answer #3: there is such a thing as a spherical pendulum like you say. When we say it has only one degree of freedom it is implicit that some sort of constraint is forcing it to remain in a plane. $\endgroup$
    – Javier
    Jan 3 '16 at 23:57
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/8860/2451 $\endgroup$
    – Qmechanic
    Jan 4 '16 at 0:09
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    $\begingroup$ To extend Javier's comment, oftentimes, textbooks cover the planar pendulum and not the spherical pendulum described by OP. $\endgroup$
    – Kyle Kanos
    Jan 4 '16 at 1:08
  • $\begingroup$ Sometimes the term is used in such a way that momentum and position are different "degrees of freedom", so then the one-dimensional oscillator would have two such "degrees of freedom". $\endgroup$ Nov 4 at 18:51
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Can you count time as a parameter?

No. The configuration is what potentially changes over time. So the equations of motion are a function from time, into the set of configurations. Time is the domain and the set of configurations is the codomain and the equations of motion is the function from the domain to the codomain.

Can't you always increase the number of parameters (even if it has no effect) and still determine the configuration of the system?

They would not be independent.

In the case of a pendulum, most texts say it has only one degree of freedom,

That's the planar pendulum, confined to rotate in a plane, like a grandfather clock.

In the case of projectile motion, the projectile has 3 degrees of freedom, right?

Yes. At each point in time you have to specify three coordinates to specify the configuration at that time. More if it is extended and can have orientation, even more if it is not rigid.

Keep in mind that a degree of freedom is about the space of possible configurations. It isn't about any one particular equation of motion.

Can someone give me a rigorous definition of the degrees of freedom and explain how this definition addresses the questions above?

The only place it seems you stumbled is about independence. You should be able to freely adjust any of the coordinates in the degree of freedom within some little bit and have a different configuration.

But this is also a false generality. For $N$ particles the degrees of freedom is $3N$ and sometimes you can pretend there are fewer by pretending that some constraint is exact when it is not actually exact. For instance a real pendulum can and does elongate a little bit and the place it pivots can wiggle a little bit and so forth. The one degree of freedom is really about ignoring the other degrees of freedom.

So just have $3N$ and then start eliminating ones you don't care about whose dynamics hardly change in an important way. And just don't over eliminate, you should retain enough to describe your system. In the case of the pendulum when you know the end point and you assume the rigidity and one part fixed, then you know the whole thing.

What can you really gain by pretending to have more generality than is really there?

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  • $\begingroup$ In the case of non-holonomic constraints, say, a rolling ball for example, there are 2 degrees of freedom, yet the number of variables required to define its configuration is more than 2, what is the definition of degrees of freedom in this case? $\endgroup$
    – Courage
    Jan 4 '16 at 5:28
  • $\begingroup$ @TheGhostOfPerdition The physics of the rolling ball will be determined by the atoms in the ball, the surface, and their interactions. The interactions might approximate a rigidity of the ball. They might even approximate a contact and a rolling. Whether you choose to approximate it as rigid is a personal choice made by the approximating human. The correct physics has $3N$ degrees of freedom, nature doesn't tell us to approximate it with 2 or more degrees of freedom. We can find the correct dynamics and we can compare that to our approximation (constrained version) and hope for a close match. $\endgroup$
    – Timaeus
    Jan 4 '16 at 5:41
  • $\begingroup$ @Courage, in the case of a non-holonomic system, the number of degrees of freedom is (n - m), where n is the number of coordinates used to describe the system and m is the number of equations that must be satisfied between them. See Whittaker's textbook at section 25 or my answer below. $\endgroup$
    – hft
    Nov 4 at 18:45
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Can someone give me a rigorous definition of the degrees of freedom and explain how this definition addresses the questions above?

Yes. A good, modern, and rigorous definition has been available for at least one hundred years. See, for example, the textbook by Whittaker titled "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies," which was written in 1917. (As an interesting aside, Dirac used this textbook for his studies of classical mechanics and their relation to quantum mechanics, in particular the definition of Poisson brackets and their relation to communtators in quantum mechanics.)

The textbook is no longer subject to copyright and is available online for free. (Google the title plus the term "pdf" to find it).

In particular, section 25 regarding "Holonomic and non-Holonomic systems" provides a definition via the following few paragraphs:

If we consider the motion of a sphere of given radius, which is constrained to move in contact with a given fixed plane, which we can take as the plane of xy, the configuration of the sphere at any instant is completely specified by five coordinates, namely the two rectangular coordinates (x, y) of the center of the sphere and the three Eulerian angles $\theta$, $\phi$, and $\psi$, which specify the orientation of the sphere about its center. The sphere can take up any position whatever, so long as it is in contact with the plane; the five coordinates (x, y, $\theta$, $\phi$, $\psi$) can therefore have any arbitrary values.

Note that in the above-quoted paragraph, "any position whatsoever" means that the sphere can slip, slide, spin in place, roll, and do any type of motion so long as a point stays in contact with the plane. This is Whittaker's case of a "perfectly smooth" plane discussed below. He contrasts this in his paragraph below with a "perfectly rough" plane where the sphere has to roll and can't slip along.

If now the plane is smooth, the displacement from any position, defined by the coordinates (x, y, $\theta$, $\phi$, $\psi$), to any adjacent position, defined by the coordinates (x + $\delta$x, y + $\delta$y, $\theta + \delta\theta$, $\phi + \delta\phi$, $\psi + \delta \psi$), where $\delta$x, $\delta$y, $\delta\theta$, $\delta\phi$, $\delta\psi$ are arbitrary independent infinitesimal quantities, is a possible displacement, i.e. the sphere can perform it without violating the constraints of the system. But if the plane is perfectly rough, this is no longer the case when$\delta$x, $\delta$y, $\delta\theta$, $\delta\phi$, $\delta\psi$ are arbitrary; for now the condition that the displacement of the point of contact is zero (to the first order of small quantities) must be satisfied, and this implies that the quantities $\delta$x, $\delta$y, $\delta\theta$, $\delta\phi$, $\delta\psi$ are arbitrary are no longer independent, but are mutually connected (in fact, they must be such as to satisfy two non-integrable linear equations); so that in the case of the sphere on the perfectly rough plane, a displacement represented by arbitrary infinitesimal changes in the coordinates is not necessarily a possible displacement.

(Emphasis in original text.)

A dynamical system for which a displacement represented by arbitrary infinitesimal changes in the coordinates is in general a possible displacement (as in the case of the sphere on the smooth plane) is said to be holonomic; when this condition is not satisfied (as in the case of the sphere on the rough plane) the system is said to be non-holonomic.

If ($\delta q_1$, $\delta q_2$, ..., $\delta q_n$) are arbitrary infinitesimal increments of the coordinates in a dynamical system, these will define a possible displacement if the system is holonomic, while for non-holonomic systems a certain number, say m, of equations must be satisfied between them in order that they may correspond to a possible displacement. The number (n — m) is called the number of degrees of freedom of the system. Holonomic systems are therefore characterized by the fact that the number of degrees of freedom is equal to the number of independent coordinates required to specify the configuration of the system.

In the last quoted paragraph above, I have added a bold emphasis to the sentence defining the term "number of degrees of freedom."

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