# Definition:Hamiltonian

## Definition

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

- $\displaystyle J\sqbrk{...y_i...}=\int_{x_0}^{x_1}\map F {x, ...y_i..., ...y_i'...}\rd x,~i=\left({1, ..., n}\right)$

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

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

## Source of Name

This entry was named for William Rowan Hamilton.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 3.13$: Derivation of the Basic Formula