p09fl28 - Lecture 28: Hamiltonians (6 Nov 09) A. Hamilton...

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Unformatted text preview: Lecture 28: Hamiltonians (6 Nov 09) A. Hamilton equations 1. In Lagrangian formulation the independent variables are (generalized) coordinates q and velocities q . In the Hamiltonian formulation, they are (generalized) coordinates q and canonical momenta p . 2. Thus calculate the total time derivative of the Lagrangian (with no explicit time dependence) L = L ( { q j } , { q j } ) dL dt = X j L q j q j + L q j q j = X j q j d dt L q j + L q j q j = d dt X j q j p j using the Lagrange equations L q j = d dt L q j = d dt p j ; p j L q j 3. Thus H j q j p j- L is conserved/constant (the energy): dH/dt = 0 4. The transformation from L to H is a Legendre transformation and the independent variables change from q , , q j to q j ,p j : dH = X j [ p j d q j + q j dp j ]- X j [ L q j dq j + L q j d q j ]- L t = X j [ q j dp j- L q j dq j ]- L t 5. Compare to the differential dH = X j [ H q j dq j + H p j dp j ] + H t 6. Then the identifications are (notice that the partials have different other coordinates held fixed for...
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p09fl28 - Lecture 28: Hamiltonians (6 Nov 09) A. Hamilton...

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