How many degrees of freedom does a spring have? I'm currently learning about thermodynamics and heat capacities. We were told that the theoretical molar heat capacities of all solids should be $3R$. I was told this is because there are 6 different vibrational degrees of freedom for each atom in a solid. Since theses atoms are stuck in place, they can't rotate and translate so I tried to figure out exactly what these 6 degrees of freedom look like. Here are my thoughts:
Firstly, I like to think of degrees of freedom as places where energy can go. For example, in a gas, energy can be transferred into a particle as kinetic energy. This kinetic energy can make it translate on each axis and rotate on each axis (assuming it's nonlinear polyatomic). Of course, each particle also has a potential energy associated with it (another place energy can go), but we can ignore it because the average potential energy of the system won't change (for fixed volume).
Now let's take a hypothetical solid that is only 1 atom thick. Its lattice will look like this:
My teacher would say this has 4 degrees of freedom because each atom in the solid can vibrate up/down or left/right. But using my definition of degrees of freedom, I would say there should only be 2 degrees of freedom. The potential energy associated with the x direction and the potential energy associated with the y direction. The reason I'm discounting kinetic energy is because when $E_{kinetic}=0$:
$$E_{mechanical} = E_{potential}$$
Therefore, we can define the amount of energy added simply by what it does to the maximum potential energy. This is why I think a normal spring only has one degree of freedom (its potential energy). So why does the system depicted above have 4 DOF's?
 A: We should start by acknowledging that "degree of freedom" has at least two meaning in physics and engineering terms. One—used in mechanical design—is essentially equivalent to a mechanical generalized coordinate for the system. Another—the more common meaning in thermal physics—is the dimensionality of the system's phase space. 
Neither of these thing is quite what you should be counting for the purposes of the equipartition theorem.

The thing that the equipartition theorem counts is contributions to the Hamiltonian that are quadratic in either a generalized coordinate or a generalized momentum.
A 1D spring has the freedom to move in only one direction (i.e. one mechanical degree of freedom using the engineering definition), but has a two dimensional phase space $(x,p_x)$ (i.e. two degrees of freedom in a common but to my ear sloppy usage), and the Hamiltonian is quadratic in both parameters
$$ H = \frac{1}{2}k x^2 + \frac{p_x^2}{2m} \,,$$
so it has two contributions for the purposes of equipartition. 
In a model solid like the one you exhibit each atom (in a $D$ dimensional space) can be naively associated with $D$ mechanical degrees of freedom, $2D$ parameters in phase space, and $2D$ quadratic modes in the Hamiltonian.

As a side note it is not necessarily true that each mechanical degree of freedom results in two contributions to the equipartition theorem. The most handy counterexample being the ideal gas.
