Category:Definitions/Hamiltonians
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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:
- $p_i = \dfrac {\partial F} {\partial y_i}$
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