**Hamiltonian mechanics** was invented in 1833 by Hamilton. Like Lagrangian mechanics, it is a re-formulation of classical mechanics.

Hamiltonian mechanics can be formulated on its own, using symplectic spaces, and not refer to any prior concepts of force or Lagrangian mechanics. See the section on its mathematical formulation for this. For the first part of this article, we will show how it has arisen historically from the study of Lagrangian mechanics.

In Lagrangian mechanics, the equations of motion are dependent on generalized coordinates {*q*_{j} | j=1,...N} and matching generalized velocities . Abusing the notation, we write the Lagrangian as , with the subscripted variables understood to represent all *N* variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as *conjugate momenta*. For each generalized velocity, there is one corresponding conjugate momentum, defined as:

- .

The *Hamiltonian* is the Legendre transform of the Lagrangian:

- .

Each side in the definition of *H* produces a differential:

- .

The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical mechanics.

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If we have a symplectic space, which comes naturally equipped with a Poisson bracket and a smooth function H over it, then H defines a one-parameter family of transformations with respect to time and this is called **Hamiltonian mechanics**. In particular, . So, if we have a probability distribution, ρ, then . This is called Liouville's theorem. Every smooth function, G, over the symplectic manifold generates a one-parameter family of symplectomorphisms and if {G,H}=0, then G is conserved and the symplectomorphisms are symmetry transformations.

See also Symplectic space.

- Weisstein, Eric W., "
*Hamiltonian*" - Rychlik, Marek, "
*Lagrangian and Hamiltonian mechanics -- A short introduction*" - Binney, James, "
*Classical Mechanics*" (PostScript) [lecture notes] (PDF)