Does any optimization problem can be stated as a "physics problem"?

Question

Is it possible to state any minimization problem as an Ising model problem or some physical problem with Hamiltonian or PDE.

Answer 1

This is not an answer, but, as a request for a clarification on the sense of the question, far too long for a comment.

Firstly, I am not entirely sure I correctly understand the meaning of „any“ in your question, I believe the English language can be rather ambiguous on this particular instance.

In other words, which of the following options is meant: A) Is there at least one minimization problem that can be stated as a “physical problem” characterised by a differential equation? B) Can all minimization problems be stated as “physical” problems?

I believe option B) is meant, as the answer to A) is trivial (the minimisation over $\mathcal {C}^1$ functions the functional $$ \int_0 ^1 (y´)^2 \mathrm{d}x$$ is equivalent to finding the static configuration of an elastic string, for example). With regards to option B), I can only share some initial thoughts. Let us consider a minimization problems expressed as functional minimization, find the function $y$ belonging to same function space that minimizes $$ \int_0 ^1 \mathcal{L} (y, y´, y´´, …) \mathrm{d}x $$ where $\mathcal{L}$ is a given function. The Euler-Lagrange condition yields a differential equation. The original question could then be translated as, can a “physical” meaning be appended to any Euler-Lagrange differential equation?

I believe a better definition of “physical” might be useful. For example, one considers the Lagrangian $$ \mathcal{L} = e^{\alpha t} (\frac{1}{2} \dot{x}^2 -\frac{1}{2} x^2 )$$ whose Euler-Lagrange equation yields the description of a damped harmonic oscillator. If $\alpha$ is negative, the system has negative friction (i.e, with a frictional force proportional to the velocity and in the same direction, contrary to “physical” damping): is it to be considered a physical system? It contradicts the Second Law, if considered in isolation: on the other hand, one could connect it with an external system, some sort of “fancy” controller, and the ODE would still model a part of a legitimate, physical situation. In this sense, it seems that given any ODE one could build a machine that provides "ad hoc" motion to a particle, by simply integrating the ODE and moving the partcile accordingly.

Similarly, if one modifies the harmonic oscillator Lagrangian $$\mathcal{L} =(\frac{1}{2} \dot{x}^2 -\frac{1}{2} x^2) \mathrm{d}x $$ by considering a “position-dependent kinetic energy”, $$\mathcal{L}_{mod} =(\frac{1}{2} V(x)\cdot \dot{x}^2 -\frac{1}{2} x^2) \mathrm{d}x $$ would you call this physical?