# Rigorously define degrees of freedom

I want to understand if there is truly a rigorous definition for the degrees of freedom in a system. Say all of a system's physical states are contained in some set $$S$$. A seemingly acceptable (and I think mostly referred to definition) is the number of real numbers needed to describe it or, more rigorously,

A system $$S$$ has $$n$$ degrees of freedom if there exists a bijective function $$f:\mathbb{R}^n\rightarrow S$$.

This however doesn't seem to uniquely define the degrees as there are bijective functions between $$\mathbb{R}^n$$ and $$\mathbb{R}^m$$ for any $$n$$ and $$m$$.

This may seem like pushing too hard into rigor but there are systems where the degrees of freedom have direct physical consequences like in statistical physics especially so I suspect there is a true and better definition out there.

My thoughts: My guess is it has something to do with topology (I am severely undertaught in this field) of the system's physical states as that is the only thing I can think of that distinguishes these different spaces (again, I have no idea). Also, this case is seemingly trivial for cases like a particle moving in 3 dimensions which of course should have 3 DOF but physical dimensions have so many more features like for e.g. scaling that can uniquely define dimension which I don't see as the case for other general systems.

• Possible duplicates: physics.stackexchange.com/q/227377/2451 , physics.stackexchange.com/q/8860/2451 and links therein. – Qmechanic Jul 26 '19 at 6:52
• @Qmechanic I think this question attacks something slightly different – Aakash Lakshmanan Jul 26 '19 at 6:54
• In this context, it’s simply the dimensikn of the configuration space manifold. – knzhou Jul 26 '19 at 7:25
• But that shifts the burden onto defining the manifold. Couldn’t you define on with lower dimensions that still maps onto all states? Sure you may have to change the topology to keep it a manifold but what determines the topology? – Aakash Lakshmanan Jul 26 '19 at 7:28
• @AakashLakshmanan the manifold is given to you with the structure of a manifold, which includes a topology. Usually the manifold is specified as a subset of ordinary n-dimensional Euclidean space R^n (because there is some set of variables with some constraints they satisfy among them in the form of equations), so the topology is inherited from R^n. The dimension, aka number of degrees of freedom, would typically be equal to the number of variables minus the number of equations they satisfy, where only “independent” equations are counted. The notion of dimension makes this more precise. – GenlyAi Jul 26 '19 at 9:04