Definition:Hamiltonian

Definition

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:

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

where:

$F$ is the Lagrangian of the system

$y_i$ are the generalized coordinates

$y_i'$ is the first derivative of $q_i$ with respect to time

$p_i := F_{y_i'}$ are the momenta of the system in those generalized coordinates:

$p_i = \dfrac {\partial F} {\partial y_i}$

Also see

• Results about Hamiltonians can be found here.

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
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hamiltonian
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hamiltonian
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hamiltonian
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Hamiltonian