Category:Definitions/Hamiltonians

This category contains definitions related to Hamiltonians.
Related results can be found in Category:Hamiltonians.

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:

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