Is there a classical analog of Bloch's theorem?

Question

In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect analogy, ODE's with periodicity in time have structure, as described by Floquet theory. Is there anything analogous for classical systems (dynamical systems) which are periodic in space rather than time? So, for example, if a system consists of a ball rolling in a periodic landscape, with a potential like $V(x,y)=\sin(x)+\sin(y)$, are there theorems that allow one to deduce anything interesting about the trajectory of the ball from the periodicity of the potential?

Answer 1

A partial answer: yes (there are such possibilities) / no (probably not in the way the OP had in mind):

The linearity of equations and the representation theory of translations are fundamental to the theories of Bloch and Floquet. In the case of Bloch, translations along a lattice vector $u$ commute with the Hamiltonian: $$ T_u \phi(x)=\psi(x+u), \; \; T_u \circ H = H\circ T_u$$ which makes possible the simultaneous diagonalisation of $T_u$ and $H$. Requiring solutions to be uniformly bounded (though not in $L^2$ globally, a small caveat) means that diagonalization of $T_u$ is done by unitary representations, thus yielding the Bloch wave-decomposition having factors of the form $e^{i \mathbf k\cdot \mathbf r}$ and its sequel.

In the case of Floquet, one considers a linear (once again) ode on a Banach space $E$, which may be written as $$\left( \frac{d}{dt} - A(t) \right) x = 0$$ When the bounded linear operaor $A$ is periodic then $A$ commutes with a time translation of period $\tau$, leaving possible the simultaneous diagonalization. This time, there being no constraint of global bounds the representations of the $T_\tau$ being of the form $M_\tau\in {\rm GL}(E)$ gives rise to the usual Floquet theory where solutions verify relations like $x(t+\tau)=M_\tau x(t)$.

When an ode is nonlinear $\dot{x}-f(t,x)=0$ but with lattice periodicity in $x$ then there is no nice commutation that comes to mind (at least mine) with consequences for solutions of the ode (if $f$ is linear and periodic in some direction, then it is constant in that direction; not so interesting). If you take e.g. the chaotic behavior of solutions to the Sinai billard (ode on a square with a convex obstacle), there seems to be little symmetry in the general solutions.

However, if you look at time-evolution of measures (or densities) under the flow of a (non-linear periodic) ode, then very likely there are Bloch/Floquet like result for eigenvalues of the now linear evolution operator. I searched failed to find a reasonable example for a flow but there are at least such results in that direction for composition maps with a lattice symmetry. An example is Floquet spectrum for weakly coupled map lattices.