The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian Mechanics, this formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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

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

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 relations between phase space variables, and without the presence of any dynamical Lagrangian or Hamiltonian, to wit,

$$ \begin{gathered} {}\{ {x^i,p_j} \} = {\delta^i_j} \\ \{ x^i,x^j \} = 0 \\ \{ p_i,p_j \} = 0 \\ \end{gathered} $$

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

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

The central equation of Hamiltonian Mechanics is the Hamilton Equation:

$$\frac{\mathrm dA}{\mathrm dt} = \{ A,H \} +\frac{\partial A}{\partial t}. $$

N.B. DO NOT USE the hamiltonian-formalism tag for Hamilton's principle. Hamilton's principle applicable to both the Lagrangian and the Hamiltonian formalisms. For Hamilton's principle use the variational-principle tag instead.