# Definition:Hamiltonian

## Definition

Let $J \sqbrk {\dotsm y_i \dotsm}$ be a functional of the form:

$\displaystyle J \sqbrk {\dotsm y_i \dotsm} = \intlimits {\int_{x_0}^{x_1} \map F {x, \cdots y_i \dotsm, \dotsm y_i \dotsm} \rd x} {i \mathop = 1} {i \mathop = n}$

Then the Hamilonian $H$ corresponding to $J \sqbrk {\dotsm y_i \dotsm}$ is defined as

$\displaystyle H = -F + \sum_{i \mathop = 1}^n y_i' F_{y_i'}$

## Source of Name

This entry was named for William Rowan Hamilton.