# p09fl32 - Lecture 32 Membranes(16 Nov 09 Homework Set VIII...

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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 VIII...

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