Tag Info

The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

# 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:

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