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.

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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

Answer

TL;DR: In classical point mechanics, (on-shell) degrees of freedom (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.