In some application of Hamiltonian Monte Carlo one can provide a full mass matrix (metric tensor) for the kinetic energy in Hamiltonian equations: $$ K(p) = \frac{1}{2}p^TM^{-1}p$$ which reduces to the correct definition from physical prespective when $M=mI$. For special relativity, however, the kinetic energy energy is $$ K(p) = m c^2 \left(\frac{p^T p}{m^2 c^2} + 1\right)^{\frac{1}{2}}$$ However, I do not see any direct way that one can have such expression for a matrix $M$.
This is of interest when we think about the Hamiltonian on a manifold. In this case the classical mass matrix is nothing more than the Riemannian metric tensor and $M^{-1}$ is the covector metric tensor (in the chosen basis). Hence, the main question is how would one write the special relativity kinetic energy on a manifold?