Given a Hamiltonian how does one find the metric it obeys? Given a Hamiltonian how does one find the metric (or families of metrics) it obeys?
For example
Let's say for the Hamiltonian $H$:
$$ H = \frac{p^2}{2m} + V(x).$$
How do I show this follows the Euclidean metric? 
 A: Sometimes it may not be possible to deduce the metric of the background a theory lives in. Consider for example Liouville theory, given by the action,
$$S = \frac{1}{4\pi}\int d^2 x \, \sqrt{g} \left(\frac{1}{2}g^{ab}\partial_a \phi \partial_b \phi + \frac{1}{\gamma} \phi \mathcal R + \frac{\mu}{2\gamma^2}e^{\gamma \phi} \right)$$
with a coupling constant $\gamma$ and cosmological constant $\mu$. This theory is invariant under conformal transformations, $g_{ab} \to \Omega(x)^2 g_{ab}$ and thus two conformally equivalent backgrounds are described by the same theory classically.
In addition, thanks to the Weyl (conformal) symmetry, we can locally set $g_{ab} = e^{\rho(x)}\delta_{ab}$, and by employing a shift in $\phi$, we can set $\rho = 0$ and thus $g_{ab} = \delta_{ab}$ resulting in the action,
$$S = \frac{1}{8\pi}\int d^2x \, \left( \partial_a \phi \partial^a \phi + \frac{\mu}{\gamma^2}e^{\gamma \phi} \right)$$
where we can omit the $\mathcal R$ term unless wanting to couple to curved space later. Thus, due to redundancies, it is not always possible to deduce the exact metric of a theory.
A: Two different metrics can have the same Hamiltonian, for instance for a conformally invariant scalar field and two conformally related metrics. Hence I do not believe it is generally possible to do. 
A: Generically, the metric is found as the coefficient of $p^2$ term. For a non-trivial metric one has $H=\frac{1}{2m} g^{ij}(x) p_i p_j$. So in your case you have $g^{ij}=\delta^{ij}$ which means that you are in the Euclidean space.
