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Considering just the Newtonian case, suppose we have a system described by $n$ canonical position-momentum pairs, $(p_1,q_1),\dots,(p_n,q_n)$, and a Hamiltonian $H$. If we "scrubbed" all the information about the $2n$ coordinates from our representation of the system, is there anything about the Hamiltonian of the system alone that uniquely determines an intrinsic number of degrees of freedom (DoF) the system has? The number of intrinsic DoF may be less than $n$. This question is essentially asking whether degrees of freedom are physically meaningful and not the result of an arbitrary choice of representation.

To be more concrete, let $\Gamma$ be the given $2n$-dimensional phase space. To "scrub" the coordinate system from our representation, define a bijection $\phi:\Gamma\to\Omega$ where $\Omega$ is a set whose elements are featureless, i.e. any two $\omega,\omega'\in\Omega$ have no information attached to them other than they are distinct nonequal elements. Then any mathematical object that takes elements of $\Gamma$ as input can be transformed to take elements of $\Omega$ as input. For example, the Hamiltonian is a function $H:\Gamma\times \mathbb{R} \to \mathbb{R}$ mapping states and time to total energy. Then the "scrubbed" version of the Hamiltonian is $H':\Omega\times\mathbb{R}\to\mathbb{R}$ where $H'(\omega,t)=H(\phi^{-1}(\omega),t)$.

We can bring over other Hamiltonian-induced information like time-evolution. Let $\tau_{\Delta t}:\Gamma\to\Gamma$ be a propagator, which is a function that time-evolves a system, i.e. $\tau_{\Delta t}(\gamma_t)=\gamma_{t+\Delta t}$ where $\gamma_t\in\Gamma$ is the state of the system at time $t$, and $\gamma_{t+\Delta t}$ is the state of the system $\Delta t$ into the future, in accordance with the Hamiltonian $H$. There is a propagator for every $\Delta t \in \mathbb{R}$, and they are uniquely determined by the given Hamiltonian. Then we have corresponding "scrubbed" propagators $\tau'_{\Delta t}:\Omega\to\Omega$.

What we cannot bring over is topological information about $\Gamma$. For example, derivatives like $\frac{\partial H}{\partial q_i}$ and $\frac{\partial H}{\partial p_i}$. We can only bring over physical observables like total energy, mass, and what state the system is in at some time (assuming distinguishability of states).

The problem is to reconstruct information about $\Gamma$ from objects like $H'$ and $\tau'_{\Delta t}$ which operate on $\Omega$ (which does not have any of the topological information about $\Gamma$ like its dimension and coordinate representation). This can include simply recovering $n$ (or perhaps a smaller number representing the minimum needed degrees of freedom).

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  • $\begingroup$ I'm not sure why you'd do this or why you would expect it to work - forgetting about topology/continuity means that you can use non-continuous bijections like $\phi : \mathbb{R}^n \to \mathbb{R}$ that then also map "n d.o.f" to a single real number. The notion of dimensionality/d.o.f. is intrinsically topological, why would you explicitly forget topological information and then ask to reconstruct it? $\endgroup$
    – ACuriousMind
    Nov 13, 2021 at 19:16
  • $\begingroup$ You might need to list all of the quantities that qualify as "physical observables" in your view. You listed some examples, but then you left the door open with the word "like," and it's not clear to me how broad that license is meant to be. Would you say that $v_i\equiv dq_i/dt$ qualifies as a physical observable? Why doesn't $\partial H/\partial p_i$ qualify? I understand that derivatives are not defined in $\Omega$, but can't we calculate the derivatives before applying the forgetful functor instead of after? And couldn't $n$ itself be considered a physical observable? $\endgroup$ Nov 13, 2021 at 19:59

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For what it's worth, even if we scrub in a less drastic way than what OP suggests and only allow smooth maps between differentiable manifolds, the question is non-trivial.

In this case, as a crude first estimate of intrinsic DOFs, we may consider the rank of the Hessian of the Hamiltonian $H$. Note that the rank may have jump/discontinuities.

Importantly, the cyclic variables lead to constants of motion, which presumably should not be considered intrinsic DOFs. Clearly, we can add canonical pairs of cyclic variables for free without changing the intrinsic dynamics of the system.

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