p09fl32 - Lecture 32: Membranes (16 Nov 09) Homework Set...

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Unformatted text preview: Lecture 32: Membranes (16 Nov 09) Homework Set VIII will be due Nov 23, not Nov 30. A. Hamiltonian theory 1. FW Sec 45. Example of constructing an equation with a current to balance a conservation law. Vector position r = x j , scalar field u . 2. Define the canonical momentum density and Hamiltonian density: P = L ( u/t ) ; H = P u t- L 3. Now with H ( r ,t ) calculate partial time derivative even though L had no explicit time-dependence (by assumption) there is a chain rule derivative for time-dependence in u and its derivatives. First the Euler- Lagrange equation with functional derivatives: t L ( u/t ) + X j x j L ( u/x j )- L u = 0 Then L t = L u u t + L ( u/t ) t u t + X j L ( u/x j ) t u x j 4. Using definition of P this gives H t =- X j x j [ L ( u/x j ) u t ] 5. The final form is expressed using a current density5....
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p09fl32 - Lecture 32: Membranes (16 Nov 09) Homework Set...

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