# What is the definition of how to count degrees of freedom?

This question resulted, rather as by-product, the discussion on how to count degrees of freedom (DOF). I extend that question here:

• Are necessary1 derivatives such as velocities counted as individual DOFs or together with the respective coordinate2?
• Should complex valued DOFs be counted twice as in "two real-valued DOFs" or once as in "one complex-valued DOF"? (I mean, when one does not want to specify this explicitly)

Please answer with some reference, unless it turns out this is actually a matter of taste rather than a strictly defined thing.

1) I mean those a value of which is required as an initial condition
2) I count fields in QFT as coordinates as well, while space-time-coordinates are parameters to me, if that matters. I know a field actually has $\infty$ (or rather, $2^\aleph$) DOFs itself, but let's say e.g. "one $\mathbb R^3$ continous DOF" in that case

TL;DR: In classical point mechanics, (on-shell) degrees of freedom$$^1$$ (DOF) are the number of initial conditions needed divided by 2.

Perhaps this is simplest to explain via an example: A 1D simple gravity pendulum with Lagrangian $$L(\theta,\dot{\theta}) = \frac{m}{2}\ell^2 \dot{\theta}^2 + mg\ell\cos(\theta)$$ has one DOF, $$\theta$$, although its solution $$\theta=\theta(t)$$ has two integration constants. Here, the generalized coordinate $$\theta$$ is the angle of the pendulum; $$\dot{\theta}$$ is the (angular) velocity; and $$p_{\theta}:=\frac{\partial L}{\partial\dot{\theta}}=m\ell^2\dot{\theta}$$ is the (angular) momentum. Furthermore, the configuration space $$M\cong\mathbb{S}^1$$ is one dimensional with one coordinate $$\theta$$. The tangent bundle $$TM$$ of the configuration space with two coordinates $$(\theta,\dot{\theta})$$ and the phase space $$T^*M$$ with two coordinates $$(\theta,p_{\theta})$$ are both two dimensional spaces. In other words, it takes two coordinates to fully describe the instantaneous state of the pendulum at a given instant $$t$$, i.e. two initial conditions.

So, to answer the main question: No, the corresponding velocity (or momentum in the Hamiltonian formulation) is not counted as a separate DOF.

For generalizations to field and gauge theory, see e.g. this Phys.SE post.

References:

1. Landau & Lifshitz, Mechanics: see e.g. first page of Chapter 1 or first page of Chapter 2;
2. H. Goldstein, Classical Mechanics: see e.g. page 13 or first page of Chapter 8 in both 2nd and 3rd edition;
3. J.V. Jose and E.J. Saletan, Classical Dynamics: A Contemporary Approach: see p.18;
4. Wikipedia, either here or here.

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$$^1$$ Exception: In the context of the equipartition theorem the DOF is conventionally the number of phase space variables that appear quadratically in the Hamiltonian.

• hm, L&L states this implicitly en passant, the wikipedia sites both lack citation, and to add confusion, the german version claims Wird ein Parameter weggelassen, so ist das System nicht mehr eindeutig bestimmt., i.e. if a parameter is removed the system is no longer uniquely determined. That would count velocity separately... Apr 19, 2011 at 8:58
• For a 1D pendulum or oscillator you have the second order equation. It can be cased into two ODE of the first order. So there are still two degrees of freedom: $\phi$ and $\dot{\phi}$. Apr 19, 2011 at 10:22
• @Vladimir: that's exactly my question (and preference). Probably it's a matter of taste (or where one studied), but to lessen confusion I guess I should stick with the definition most-used in the literature - it's hard for me to question L&L or Goldstein there :-/ edit maybe it's the difference between a mathematical view ($\dot\phi$ is independent of $\phi$) vs an experimental view ($\dot\phi$ just describes the change of $\phi$ but not another particle/field/...) Apr 19, 2011 at 10:27
• regardless, the physical fact of the 1D pendulum example is that you only need a single scalar variable to describe its state (when it is in its steady state of course!). any other introduced variables are not separate. Apr 19, 2011 at 12:37
• @Bjorn: The configuration space of the 1-D pendulum is clearly 2-D, by which I mean that with both the angle and angular velocity specified I understand the whole past and future behavior of the system. I suspect that what is happening here is that we use "degrees of freedom" in two ways. One way describes the mechanical constrains on a system (i.e. a pendulum swings on one axis only), and the other way counts the dimensionality of the configuration space. Qmechanic's references support the first meaning. I'll be interested in seeing references that use the second. Apr 19, 2011 at 17:06