What is Degrees of Freedom in Physics?
Definition
The number of independent coordinates among the total coordinates of a particle system is called the degree of freedom.
Explanation
To put it simply, the degree of freedom is the minimum number of variables needed to describe a particle system. Consider a particle moving freely in three-dimensional space. The position of this particle can be expressed as $r = (x,y,z)$, and since the variables for each axis $x, y, z$ are independent of each other, the degree of freedom of this particle system is $3$.
As in the example below, if specific conditions are added to the movement of the particles, the degree of freedom is reduced. These conditions are called constraint conditions. Furthermore, a particle system with $n$ degree of freedom represented by $n$ coordinates is called generalized coordinates.
Circular Motion
Consider a particle moving in a circular motion along a unit circle in two-dimensional space. The position of the particle can be expressed as $r = (x,y)$. In this case, since $y=\sqrt{1-x^{2}}$, the degree of freedom of this particle system is $2-1=1$.
$$ \begin{align*} x &= x\\ y &= \sqrt{1 - x^{2}} \end{align*} \quad \implies \quad r = (x, \sqrt{1-x^{2}}) $$
Of course, in this case, it is much more convenient to express it with the angle $\theta$. $$ r = (\cos\theta, \sin\theta) $$ Here, the constraint condition is $x^{2} + y^{2} = 1$, and the generalized coordinates are $\theta$.
Double Pendulum
Consider a double pendulum with a radius of $R, r (R \gt r)$. To represent the positions of the two pendulums in two dimensions, 4 coordinates $(x_{1}, y_{1})$, $(x_{2}, y_{2})$ are needed, but the degree of freedom is $2$. If the angles formed with the $x$ axis by the two pendulums are called $\theta_{1}, \theta_{2}$, then all positions of the particle system can be expressed with these two variables.
$$ \begin{align*} x_{1} &= R\cos\theta_{1} \\ y_{1} &= R\sin\theta_{1} \\ x_{2} &= R\cos\theta_{1} + r\cos\theta_{2} \\ y_{2} &= R\sin\theta_{1} + r\sin\theta_{2} \end{align*} $$
Here, the generalized coordinates are $(\theta_{1}, \theta_{2})$.
