\begin{definition}[Definition:Hamiltonian]
Let $J \sqbrk {\dotsm y_i \dotsm}$ be a functional of the form:
:$\ds J \sqbrk {\dotsm y_i \dotsm} = \intlimits {\int_{x_0}^{x_1} \map F {x, \cdots y_i \dotsm, \dotsm y_i \dotsm} \rd x} {i \mathop = 1} {i \mathop = n}$
Then the '''Hamiltonian''' $H$ corresponding to $J \sqbrk {\dotsm y_i \dotsm}$ is defined as
:$\ds H = -F + \sum_{i \mathop = 1}^n y_i' F_{y_i'}$
{{NamedforDef|William Rowan Hamilton|cat = Hamilton}}
\end{definition}