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