Hamiltonian mechanics really useful for numerical integration? Lagrange equations can become 1st-order by introducing extra variables
Comments to the question (v8) concerning numerical integration:
On one hand, to solve a Hamiltonian system numerically, there exist the numerical integration schemes of symplectic integrators (SI), where each (finite) numerical iteration step is a canonical transformation/symplectomorphism, which preserves certain properties, such as, e.g., energy, and which makes SIs suitable to solve long-term evolution problems numerically.
On the other hand, transforming a 2nd-order coupled ODE system $$\ddot{q}^i~=~f^i(q,\dot{q},t)\tag{1}$$ into a 1st-order coupled ODE system $$\dot{v}^i~=~f^i(q,v,t), \qquad \dot{q}^i~=~v^i,\tag{2}$$ does not necessarily bring it on Hamiltonian form$^1$.
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$^1$ If the Lagrangian is of the form $L(q,v,t)=\frac{1}{2}mv^2-V(q,t)$ so that momentum $p=mv$ is proportional to velocity, then the 1st-order system (2) is on Hamiltonian form. See also e.g. this Wikipedia page.