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The Hamiltonian

The Hamiltonian can be interpreted as an “energy input”, as opposed to a Lagrangian, which is the "energy output". 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 relationships and the Dynamic Hamiltonian Relationships

The Poisson Bracket relations are algebraic relationships between phase space variables, and without the presence of any dynamical Lagrangian or Hamiltonian. Thus, the Poisson Bracket relations would obviously (to someone with a basic knowledge of Lagrangian Mechanics) be :

$$ \begin{gathered} \{ {{p_i},{x_j}} \} = {\delta _{ij}} \\ \{ {{p_i},{p_j}} \} = 0 \\ \{ {{x_i},{x_j}} \} = 0 \\ \end{gathered} $$

The Dynamical Relationships, however, are obviously changed. It is clear that the new relationshipjs are that:

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

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:

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

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