What is degree of freedom?

Question

My major was physics before i transferred my major to mathematics, so i'm quite familiar with this term "degree of freedom". (And i'm a junior now)

However, even when i was majoring physics, i was not sure what's exactly the definition of this.

I was reading an article in wikipedia about Euclidean group and the term degree of freedom appeared here: en.m.wikipedia.org/wiki/Euclidean_group

I searched the page linked on the term, but there was no rigorous definition of it..

What is the definition of degree of freedom formally?

Answer 1

"Degrees of freedom" takes on a slightly different meaning in different contexts, but in the arena of pure mathematics, this comes down to "the number of parameters of the system that may vary independently". In general, I tend to think of it as "the minimum number of pieces of information required to describe an element uniquely".

Usually, this notion is equivalent to that of dimension, but using the same term for both makes it unclear whether we are talking about the space itself (i.e. 2-dimensional or 3-dimensional space) or the system within it (i.e. the Euclidean group, with its 6 "dimensions", i.e. its 6 degrees of freedom).

In the context of the Euclidean group of $3$ dimensional space, the statement that there are $6$ degrees of freedom means that it would take $6$ real numbers to completely describe any member of the group, which in this case means an isometry. We would need $3$ numbers to determine the coordinates by which the object is translated, and $3$ numbers to determine the associated orthogonal transformation.

Answer 2

For me, a "state" of a "system" in physics is described by a point in a manifold and the number of degree of freedom is the dimension of the manifold.

(and the "system" is the algebra of functions on that manifold.)

An idea contained in the definition of manifold of dimension n is that the position of a point is given by n numbers, similarly to a vector in a vector space, once a basis is given. However there is no addition.

Answer 3

A degree of freedom is the ability to move along, or rotate about an axis in the given room.

So for $\mathbf{R}$3, being able to move and rotate in all directions yields six degrees of freedom.