Degree of Freedom

Definition: Degree of Freedom

Given a parameterisation of a physical system, the number of degrees of freedom is the smallest number $n$ of parameters whose values need to be known in order to always be able to determine the values of all parameters in the chosen parameterisation.

In this case, any set of $n$ such parameters are called degrees of freedom.

EXAMPLE

If we parameterise a molecule floating in free space only by position, i.e. its $x,y,z$ coordinates In this parameterisation, we have $3$ degrees of freedom, since $3$ is the smallest number of parameters needed to determine all parameters - you cannot use a single coordinate to calculate the others and you cannot use just $2$ coordinates to calculate the third.

Similarly, if we parameterise it only by its velocity, then its velocity components $v_x,v_y,v_z$ are degrees of freedom since they are independent of each other.

EXAMPLE

One possible parameterisation of a gas is through its pressure $p$, temperature $T$, volume $V$ and number of particles $N$. It can be proven that the gas satisfies the following law, where $k$ is a known constant:

$$pV=kNT$$

Even though we have $4$ parameters - $p,V,T,N$ - we have only $3$ degrees of freedom, since the above law can be used to determine the value of the $4$-th parameter if we know the values of the other three.