The Hamiltonian

The Euclidean Hamiltonian, which is used in Classical Mechanics is given by:

$$H = \frac{p^2}{2m} + U$$

The Euclidean Lagrangian, on the other hand, has a minus instead of a plus.

Notice that

$$L + H = p\frac{\text{d}x}{\text{d}t}$$

This shows that the two are related by a Legendre transformation.

The Poisson Bracket relations and the Dynamic Hamiltonian relations

The Poisson Bracket relations are algebraic relationships between phase space variables, and without the presence of any dynamical Lagrangian or Hamiltonian; they read $$ \begin{gathered} \{ {{p_i},{x_j}} \} = {\delta _{ij}} \\ \{ {{p_i},{p_j}} \} = 0 \\ \{ {{x_i},{x_j}} \} = 0 \\ \end{gathered} $$

The dynamical relations are

$$\begin{gathered} \frac{{\partial H}}{{\partial {\mathbf{x}}}} = - \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}t}} \\ \frac{{\partial H}}{{\partial {\mathbf{p}}}} = \frac{{{\text{d}}{\mathbf{x}}}}{{{\text{d}}t}} \\ \end{gathered} $$

Compare this to the Dynamical Lagrangian Relations:

\begin{gathered} \frac{{\partial L}}{{\partial {\mathbf{x}}}} = \frac{{{\text{d}}{\mathbf{p}}}}{{{\text{d}}t}} \\ \frac{{\partial L}}{{\partial {\mathbf{p}}}} = \frac{{{\text{d}}{\mathbf{x}}}}{{{\text{d}}t}} \\ \end{gathered}

The central equation of Hamiltonian Mechanics is the Hamilton Equation:

$$\frac{{{\text{d}}A}}{{{\text{d}}t}} = \{A,H \} $$