= Solution
The <canonical momentum> is $p_i=\partial L/\partial\dot q_i$. If the velocity-momentum relation can be locally inverted, the <Legendre transform> defines $H(q,p,t)=\sum_ip_i\dot q_i-L$, with the velocities expressed in terms of $q,p,t$. Its differential is
$$
dH=\sum_i\dot q_i\,dp_i-\sum_i\frac{\partial L}{\partial q_i}dq_i-\frac{\partial L}{\partial t}dt.
$$
The <Euler-Lagrange equations> then yield <Hamilton's equations>,
$$
\boxed{\dot q_i=\frac{\partial H}{\partial p_i},\qquad\dot p_i=-\frac{\partial H}{\partial q_i}.}
$$
A singular velocity Hessian instead requires a constrained-Hamiltonian treatment.
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