= Solution
The <Poisson bracket> is antisymmetric, so $\{H,H\}=0$. The <Hamiltonian energy balance> consequently gives
$$
\boxed{\frac{dH}{dt}=\frac{\partial H}{\partial t}=0.}
$$
For a <regular Lagrangian> the <Legendre transform in mechanics> gives $H_t=-L_t$, evaluated at the corresponding velocity. In particular, a <Lagrangian> with no explicit time dependence has <time-translation symmetry>. <Noether theorem> associates this continuous <symmetry> of the <action> with the <conserved quantity> $\sum_i\dot q_i\partial L/\partial\dot q_i-L=H$. For an ordinary natural mechanical <Lagrangian> $L=T-V$, with $T$ quadratic in the velocities, this quantity is the total <energy> $T+V$.
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