Formalism of $H=E$ (Hamiltonian mechanics)

The function $h$ defined by the equation $$h=sum_jleft( frac{partial mathcal{L}}{partial dot{q}_j}dot{q}_jright)-mathcal{L}$$ is called the energy function of the system $mathcal{S}$.

If $mathcal{L}=mathcal{L}(q,dot{q},t)$, then $partial mathcal{L}/partial tnot=0$ and $h$ is not conserved.

If $mathcal{L}=mathcal{L}(q,dot{q})$ then $partial mathcal{L}/partial t =0$ so that $h$ is a constant. The conservation formula $$sum_jleft( frac{partial mathcal{L}}{partial dot{q}_j}dot{q}_jright)-mathcal{L}=mathrm{constant}$$ is called the energy integral of the system $mathcal{S}$.

If $mathcal{S}$ is a conservative standard system, then $mathcal{S}$ is autonomous and so $h$ is conserved. In addition, energy integral can be written in a more familiar form, if

$$T=sum_{j,k}a_{jk}(q)dot{q}_jdot{q}_k$$

and $V=V(q)$. The energy integral becomes $$h=T+V=mathrm{constant}$$ In this case the total energy $E$ of the system.

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Now All you need to do is to take Hamiltonian in place of the Energy function. $$mathcal{H}=sum_jleft( frac{partial mathcal{L}}{partial dot{q}_j}dot{q}_jright)-mathcal{L}=sum_j p_jdot{q}_j-mathcal{L}$$