Classical analogue of Heisenberg and Schrödinger pictures? What do the Heisenberg and Schrödinger pictures in quantum mechanics correspond to in classical mechanics (if they correspond to anything)? It's kind of weird, because (if I understand it well) in classical mechanics the state of the system is determined by the dynamical variables positions and conjugate momenta, so the time dependence of the dynamical variables is the same as the time dependence of the state vector, I don't see whether the two could be separated. Nevertheless can it be done somehow?
 A: There is really no analogue of the "pure state" or "state vector" in classical physics so there is no analogue of Schödinger's equation for ket vectors.
However, the density matrix has an analogue in classical physics, namely the probability distribution on the phase space, also called $\rho$. Of course, when we talk about it, we assume a statistical description i.e. we assume some ignorant about the classical system. The classical counterpart (and the classical limit!) of the "Schrödinger equation for the density matrix" is called the Liouville equation or the equation from the Liouville's theorem, so that it's not confused with other things called the "Liouville equation".
The quantum counterpart of the usual dynamical equations from classical physics are of course the Heisenberg equations of motion, the equations describing the evolution in the Heisenberg picture. The classical equations of motion are literally the classical, $\hbar\to 0$, limit of the Heisenberg equations of motion.
I have oversimplified a bit. There is no classical counterpart of the wave function among the things that we normally learn about. However, there exists the Koopman-vonNeumann classical mechanics

https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics

that is doing nothing else than to construct some "wave function"-based description of classical physics, including Born's rule.
A: From The Gateless Gate by Mumon,

Two monks were arguing about a flag. One said: "The flag is moving."
The other said: "The wind is moving."
The sixth patriarch happened to be passing by. He told them: "Not the
  wind, not the flag; mind is moving."

The Heisenberg Picture is the the first monk, the Schroedinger picture is the second monk.
The approaches are mathematically equivalent, you have some state space manifold and you define some time flow along the manifold. The two pictures are dual to each other, like Lagrangian and Hamiltonian mechanics (on the tangent and cotangent bundles respectively). Have a look at this paper for a more complete explanation.
A: The already posted answer is, strictly speaking, correct; however i will try to provide an analogy of Heisenberg/Schrödinger pictures which can be classical.
The difference between the pictures is what part of the description carries the time dependence and evolution.
Heisenberg picture, has (the operators of) the observables carry the time-evolution while the state is not affected by time evolution.
In Schrödinger picture, the state carries the time evolution, while any operator-observables are not affected by it.
Now, consider a (single particle) classical system which is evolving in some frame of reference. One can assume a picture of the system where the frame of reference is static (lets say this is "Heisenberg picture of the system") and the particle observables (position/momentum) are evolving (wrt to the frame of ref). Or, on the other hand, say the system observables are static (wrt the frame of ref) while the frame itself is evolving (lets say this is "Schrödinger picture of the system"). This can be generalised to multiple particle systems, but is good for an analogy.
If one wants a closer analogy, one can assume a system which is liquid or gas and the frame of reference can be curvlinear. This will make a probabilistic / density description possible and is closer to a "quantum-like description"
A: Measurable quantities are described by time evolution of observables classically. Think about this for a bit. In Classical Mechanics what we see evolving in time are quantities like momentum, position, etc, i.e observable quantities. This is why the Heisenberg picture is useful. Exactly the same expectation value is obtained when we leave the state ﬁxed as $|\psi(0)\rangle$ and allow the observable to evolve in time as $U(t)^†\hat{O}U(t)$ (Heisenberg picture) as opposed to the Schrodinger picture which is the state that evolves in time. 
The equations of motion obtained from the Heisenberg picture look familiar from the equivalent ones in Classical Mechanics hence we can better demonstrate similarities (or differences) between QM and CM. Historically, the Heisenberg picture was developed (by Heisenberg) for his matrix mechanics formulation of quantum theory, and was developed in parallel to the wave-function formulation of quantum mechanics.
As for the Schrodinger picture, there is no classical analogue that I know of, and its one of the reasons that we shift constantly from picture to picture (Schrodinger, Heisenberg, Interaction pictures) when solving problems.
A: Not a general answer but an instance related to this is provided by the fluctuation dissipation theorem for time-dependent expectation values of an observable in statistical mechanics, close to equilibrium we have the relation
\begin{equation}
  \frac{\langle\theta_A\rangle_{neq}(t)-\langle\theta_A\rangle_{eq}}{\langle\theta_A\rangle_{neq}(0)-\langle\theta_A\rangle_{eq}}=\frac{\langle\theta_A(0)\theta_A(t)\rangle_{eq}-\langle\theta_A\rangle_{eq}^2}{\langle \theta_A^2\rangle_{eq}-\langle\theta_A\rangle_{eq}^2}
\end{equation}
here we are able to translate between a time-dependent distribution function $f(q_0,p_0,t)$ (Schrodinger)
$$
\langle\theta_A\rangle_{neq}=\int_{-\infty}^{\infty}dq_0 \int_{-\infty}^{\infty} dp_0  \quad \theta(q_0,p_0) f(q_0,p_0,t)
$$
and a time-dependant observable (Heisenberg)
$$
\langle\theta_A(0)\theta_A(t)\rangle_{eq}=\int_{-\infty}^{\infty}dq_0 \int_{-\infty}^{\infty} dp_0  \quad \theta(q_0,p_0)\theta(q(t),p(t)) e^{-H(q_0,p_0)/k_B T} 
$$
It would be interesting to know if one can do this more generally
