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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
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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? $\endgroup$ – Oswald 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

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