p09fl27 - Lecture 27: Hamiltons equations (4 Nov 09) Review...

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Unformatted text preview: Lecture 27: Hamiltons equations (4 Nov 09) Review 2 Nov homework A. Review Hamiltonian 1. FW Secs. 18 and 20 and L11 2. In the case that the Lagrangian has no explicit time dependence, L = L ( { q j } , { q j } ), form the total time derivative, with variation through the q j and q j : dL dt = X j L q j q j + L q j q j 3. Lagrange equations give L q j = d dt L q j = d dt p j ; p j L q j 4. so dL dt = X j q j d dt L q j + L q j q j = d dt X j q j p j 5. Thus H j q j p j- L is a conserved quantity (the energy): dH dt = 0 6. Lorentz force from L10 (Sec. 33 of FW). The Lagrangian that generates the equation of motion for particle of mass m and charge q in electric and magnetic fields E and B (potential and vector potential and A ) is L = m 2 v 2- q + q v A The canonical momentum is then ( v = r ) p = m v + q A 1 (mechanical momentum m v ) and the Hamiltonian is H = v p- m 2...
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This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at Wisconsin.

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p09fl27 - Lecture 27: Hamiltons equations (4 Nov 09) Review...

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