What does degrees of freedom mean in the context of vibrations?

Question

1. If you have an $N$ degrees of freedom system what does this mean?

2. What is the difference between a 1 and a 2 degrees of freedom system?

Answer 1

For example an object can vibrate in one dimension only (e.g $x$, thus $1$ degree of freedom).

Or an object can vibrate in 2 dimensions (e.g $x$ and $y$, $2$ degrees of freedom)

Furthermore an object can vibrate in a rotational sense, a further degree of freedom (in this case an angle lets say $\phi$, in a classical sense and not a quantum-mechanic sense).

References:

1. http://en.wikipedia.org/wiki/Degrees_of_freedom_%28physics_and_chemistry%29

2. http://en.wikipedia.org/wiki/Normal_mode

3. http://en.wikipedia.org/wiki/Rotation

4. http://en.wikipedia.org/wiki/Molecular_vibration

Answer 2

One degree of freedom is a straight line between 2 points. It has no width and no plane in which to vibrate. A line between 2 points involves distance which implies time. Assuming that time is a dimension a straight line requires time and a 2 dimensional plane in which to vibrate. Hence, three dimensions are required to allow the simplest of vibrations. Four dimensions allow a simple vibration to rotate, and so on up. String theory allows more complex wave functions in higher dimensions, each requiring their own manifolds (higher dimensional spaces).