p09fl29 - Lecture 29: Hamilton-Jacobi Poisson brackets (9...

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Unformatted text preview: Lecture 29: Hamilton-Jacobi Poisson brackets (9 Nov 09) A. Canonical transformations 1. FW Secs. 34,35; LL Secs. 46, 47 2. Transform to new coordinates Q j , P j p j = p j ( P i ,Q i ,t ); q j = q j ( P i ,Q i ,t ) 3. Ask for transformations that keep the form of Hamiltons equations for the new Hamiltonian H : Q j = H P j ; P j =- H Q j 4. This will be the situation if the modified Hamiltons principle is avail- able for the new coordinates. That allows a bit of generality: X j q j p j- H = X j Q j P j- H + dF 1 ( q,Q,t ) dt p j = F 1 q j ; P j =- F 1 Q j ; H ( Q,P,t ) = H ( p,q,t ) + F 1 t 5. For Hamilton-Jacobi theory, one changes the variables in the added function F to q,P by the Legendre transformation F 2 = X j Q j P j + F 1 X j q j p j- H = X j Q j P j- H + dF 1 ( q,Q,t ) dt =- X j Q j P j- H + dF 2 ( q,P,t ) dt Then set up the total time derivative of F 2 : dF 2 dt = X j F 2 q j q j + F...
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p09fl29 - Lecture 29: Hamilton-Jacobi Poisson brackets (9...

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